Method and apparatus for providing improved human observer xyz functions and calculations for cielab

ABSTRACT

A method for determining color-matching functions includes obtaining spectral data originating from metameric pairs and generating new color matching functions by modifying original color matching functions. The new color matching functions are constrained to be similar to the original color matching functions while reducing calculated perceptual error between the metameric pairs. In another embodiment a method for determining color-matching functions spectral data originating from color matching experiments and from metameric pairs are generated. Color matching functions from a set of parameters and an error function that indicates error due to perceptual differences between the parameterized color matching functions and the color matching experiments are defined. Methods for defining human observer functions, optimizing the definitions of LMS cone response functions as well as other methods and systems are also disclosed.

FIELD OF THE INVENTION

This invention relates in general to the field of color measurement andin particular to a method and apparatus for optimizing the LMS coneresponses which can be used as the basis for the XYZ human observerfunctions in order to better predict metameric matches using publishedcolor matching functions in conjunction with sets of spectral datacomprising of metameric pairs.

BACKGROUND OF THE INVENTION

The human experience of color is dependent on the spectral properties ofthe stimulant, i.e. the color being observed, combined with the spectralsensitivities of the eye+brain system. This in turn is governed by thespectral sensitivities of the LMS (long, medium, short wavelength) conesresiding in the retina and the neural and cognitive processing of theLMS responses performed by the retina and brain. Thus, whereas measuringthe spectra of a color stimulant is objective and unambiguous, measuringthe human sensation known as color is dependent on the validity of whatare referred to as the human observer functions CIEXYZ (which definecolor matching and can be related to LMS) and the calculation for CIELAB(which is dependent on XYZ and defines how colors are perceived).

Since CIEXYZ and CIELAB are derived from experiments involving humanobservers, the validity of CIEXYZ and CIELAB may be compromised byerrors in the experimental data acquired (in the case of color matchingexperiments), errors in the methods used to convert the raw data tohuman observer functions, and errors in color order systems such asMunsell which were used to optimize the calculation for CIELAB.

The imperfections in CIEXYZ and CIELAB are well-known in the industry:

-   -   a) Colors that measure the same values of CIEXYZ and CIELAB do        not necessarily match, especially whites comprised of        significantly different spectra.    -   b) Colors that differ by the same Euclidean distance in CIELAB        may appear nearly identical or may appear dramatically different        depending on the location of the colors within the CIELAB        coordinate system and the direction of their color difference.

If the current methods for calculating CIEXYZ and CIELAB were valid,colors with significantly different spectra but similar values of CIEXYZand CIELAB would match visually. Such pairs of colors are called“metameric pairs” or “metamers”. To the extant that such pairs of colorsdo not match visually, one can say that CIEXYZ and CIELAB are imperfectmethods for defining or measuring color. This results in pairs of colorsthat are metameric matches numerically but not visually.

In order to identify possible sources of error in the definitions ofCIEXYZ and CIELAB, we begin with a very brief history of modern daycolorimetry. The Commission Internationale de l'Éclairage (hereinafter“CIE”) XYZ observer functions are the basis for most color measurementsthat require the matching of colors. By combining these functions withnon-linear color appearance models (CAMs) such as CIELAB and CIECAM96sCAMs, complex color images as well as simple color patches can bereproduced with great success if the color media are similar in spectralbehavior.

The color matching functions of the CIE 1931 standard were based on thedata of Guild using seven observers and Wright using ten observerstogether with the CIE 1924 luminous efficiency function V (λ). As aresult of visual vs. numerical discrepancies in the matching of paperwhite, Stiles performed a “pilot” repeat of the 1931 determination ofthe color matching functions using 10 observers and (together withBurch) a “final” version using 49 observers in 1958. This latter “final”experiment was performed using a larger field of view (10 degree ratherthan 2 degree) and hence the two standards from 1931 and 1958 arereferred to as the 2 and 10-degree observer, respectively. Example plotsof the raw color matching data acquired from the multiple observers byStiles and Burch clearly indicate significant noise and variability inthe data, probably due to the procedures used to obtain the colormatches and the imperfect skill of the participants in the art ofadjusting colors for purposes of obtaining a match.

In his book “The Reproduction of Colour”, Robert Hunt relates thehistorical XYZ observer functions to the spectral sensitivities of theLMS cones in the retina. Hunt cites Estevez and similar referencesregarding the red, green, blue spectral sensitivities of the cones (p.706). By comparing these estimates with the existing XYZ observerfunctions, Hunt defines the Hunt-Estevez-Pointer conversion thatconverts the XYZ observer functions to the long-medium-short spectralsensitivities of the eye LMS (as defined in most references) orequivalently ρ, β, γ(the labels Hunt prefers to avoid confusion withother color values such as using “L” for luminance). TheHunt-Estevez-Pointer matrix is defined as:

$\begin{matrix}{M_{{XYZ}->{LMS}} = \begin{pmatrix}0.38971 & 0.68898 & {- 0.07868} \\{- 0.22981} & 1.18340 & 0.04641 \\0 & 0 & 1.00\end{pmatrix}} & {{Eq}.\mspace{14mu} \left( {1\text{-}1} \right)}\end{matrix}$

While the CIEXYZ observer functions appear to work well between mediathat are somewhat similar using similar lighting conditions, thereappears to be a discrepancy between measurement and visual matches formedia with radically different spectra, particularly in the areas ofwhites. This discrepancy has been documented as the impetus for the 10degree (vs. 2 degree) observer function effort performed by Stiles andBurch from 1955-1958 and is the basis for the deviation from standardCIE XYZ metrology developed in conjunction with MATCHPRINT™ Virtualtechnology from Eastman Kodak, which is used in a variety of systemsthat require an accurate visual color match between displays and printedimages viewed under standard illumination.

The methods used to improve the CIE functions fall into two categories.The first category are methods such as those used by Thornton (1998) andalso by Matsushiro, Ohta, Shaw, and Fairchild (2001) which attempt toalter the human observer functions on a wavelength by wavelength basis,possibly with constraints, in order to minimize error between metamericpairs as well as to remain similar to the original human observerfunctions. This effectively allows the number of adjustable parametersto be 3×32=96 or higher.

The second category of approach is to use a small number of parametersas discussed in Fairchild (1989) and North and Fairchild (1993) relyingfor example on the “deviant observer” which attempts to account foryellowing effects of aging due to the lens and the macula by applyingbroad band absorption spectra to the LMS functions. This approach hasthe deficiency of being very limited in how it modifies the standardhuman observer functions. So although this approach can accurately mimicthe standard human observer functions, it cannot adequately modify thefunctions in order to reduce the calculated ΔE errors between metamericpairs that have significantly different spectra and yet match visually.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a plot for LMS in accordance with one embodiment of theinvention.

FIG. 2 shows a graph highlighting color-matching function predictedversus data at full scale for R in accordance with an embodiment of theinvention.

FIG. 3 shows a graph highlighting color-matching function predictedversus data magnified ×25 for G and B in accordance with an embodimentof the invention.

FIG. 4 shows a graph highlighting actual XYZ observer functions versusresults of calculating LMS and converting to XYZ in accordance with anembodiment of the invention.

FIG. 5 shows a graph highlighting plots for CIEXYZ versus EdgeXYZ inaccordance with an embodiment of the invention.

FIG. 6 shows a graph highlighting plots for EdgeXYZ versus CIEXYZ for 10degree observer functions in accordance with an embodiment of theinvention.

FIG. 7 shows a graph highlighting a plot of EdgeXYZ model for 10 degreeobservers to compare with 2 degree standard with similar whitenormalization in accordance with an embodiment of the invention.

FIG. 8 shows a graph highlighting a plot of EdgeXYZ versus the 2 degreestandard observer in accordance with an embodiment of the invention.

FIG. 9 shows a plot of EdgeXYZ versus CIEXYZ (2 degree observer) inaccordance with an embodiment of the invention.

FIG. 10 shows a graph highlighting plots for s(λ) in accordance with anembodiment of the invention.

FIG. 11 shows a block diagram of a circuit that models a human observerin accordance with an embodiment of the invention.

FIG. 12 shows a block diagram of a system in accordance with anembodiment of the invention.

FIG. 13 shows a graph highlighting a comparison of the current inventionXYZ versus CIELAB XYZ in accordance with an embodiment of the invention.

FIG. 14 shows a device in accordance with an embodiment of theinvention.

FIG. 15 shows a flow chart for a technique for determining colormatching functions in accordance with an embodiment of the invention.

FIG. 16 shows a flow chart for a technique for determining colormatching functions in accordance with another embodiment of theinvention.

DETAILED DESCRIPTION OF THE INVENTION

Experiments involving metameric pairs can be used to assess the validityof existing methods for calculating CIEXYZ and CIELAB. The results ofsuch experiments can be used by this invention to improve CIEXYZ.Likewise, comparing the relative differences of colors defined in acolor order system such as Munsell can assess the validity of bothCIEXYZ and CIELAB with regard to perceptual uniformity of thesenumerical definitions of color. The results of such comparisons can beused to improve CIEXYZ and CIELAB.

However, in the event that the color order system itself is flawed ormisinterpreted regarding its use in validating CIEXYZ and CIELAB,independent tests can be devised to confirm whether the color ordersystem itself is adequate for optimizing the calculation used fordefining color measurement. These independent tests can comprise of verysimple test images that display steps of color that change in aprogression by a consistent increment as calculated by CIELAB, both indifferent regions of color, and in various directions of color. In theevent that the color order system is inadequate for purposes ofoptimizing CIELAB, the test images will displays progressions of colorchange that are dramatic in some color directions and in certain regionsof color, and are barely noticeable in other color directions and/orregions of colors.

By visually editing and optimizing such test images such that theseprogressions are visually consistent, one can optimize CIEXYZ and CIELABby means of the measured data from these charts, treating the data setas a subset of a new color order system. Since embodiments of theinvention is based upon a new understanding of the physics of humancolor vision, it is possible to optimize CIEXYZ and CIELAB in aneffective manner that results in a successful validation test using theabovementioned charts, and which will be valid even for a color ordersystem comprising of many samples. In fact, a new color order system cannow be defined that is based solely on equal increments in this improvedCIEXYZ and CIELAB space, which we will be refer hereinafter as “EdgeXYZ”and “EdgeLAB”.

In the following description a method and apparatus are described foroptimizing the LMS cone responses which can be used as the basis for theXYZ human observer functions using published color matching functions inconjunction with sets of spectral data comprising of metameric pairs.The method and apparatus are further described regarding theoptimization of the conversion of LMS to XYZ and the coefficients usedto calculate CIELAB using data from color order systems and fromvalidation charts for testing perceptual uniformity of CIELAB. Oneobjective of the invention is to use far fewer parameters than the firstcategory previously described which attempt to alter the human observerfunctions on a wavelength-by-wavelength basis, possibly withconstraints, but yet still be able to lower the calculated ΔE errorsbetween metameric pairs by a fairly large amount (e.g., at least 50%).

The description begins by constructing a framework within which we canclarify the mathematics of color matching experiments, such as wasperformed by Guild and Wright. The following notation is used: the{right arrow over (λ)}_(T) vector refers to the choice of tristimulouswavelengths used to match a color of wavelength λ. The components of thevector {right arrow over (RGB)}(λ,{right arrow over (λ)}_(T)) are themagnitudes of the tristimulous wavelengths required to match the colorof wavelength λ. Note that negative values of any of the threecomponents of {right arrow over (RGB)}(λ,{right arrow over (λ)}_(T))denote adding a quantity of light given by the magnitude of thatcomponent (the wavelength of the light being the same as that of thecomponent) to the color of wavelength λ that is being matched.

These components are by definition the CMFs for the particulartristimulous wavelengths {right arrow over (λ)}_(T). In a similarmanner, vector functions for XYZ and LMS are defined that are calculatedfrom the Dirac delta function that are used to characterize the spectrapower distribution (SPD) of a monochromatic light source of wavelengthλ, together with the human observer functions x(λ),y(λ),z(λ):

$\begin{matrix}\begin{matrix}{{\overset{->}{\lambda}}_{T} = \begin{pmatrix}\lambda_{r} \\\lambda_{g} \\\lambda_{b}\end{pmatrix}} \\{{\overset{}{RGB}\left( {\lambda,{\overset{->}{\lambda}}_{T}} \right)} = \begin{pmatrix}{R\left( {\lambda,{\overset{->}{\lambda}}_{T}} \right)} \\{G\left( {\lambda,{\overset{->}{\lambda}}_{T}} \right)} \\{B\left( {\lambda,{\overset{->}{\lambda}}_{T}} \right)}\end{pmatrix}} \\{{\overset{}{XYZ}(\lambda)} = \begin{pmatrix}{X(\lambda)} \\{Y(\lambda)} \\{Z(\lambda)}\end{pmatrix}} \\{{\overset{}{LMS}(\lambda)} = \begin{pmatrix}{L(\lambda)} \\{M(\lambda)} \\{S(\lambda)}\end{pmatrix}}\end{matrix} & {{Eq}.\mspace{11mu} \left( {1\text{-}2} \right)} \\\begin{matrix}{{S_{\lambda}\left( \lambda^{\prime} \right)} = {\delta \left( {\lambda^{\prime} - \lambda} \right)}} \\{{\overset{}{XYZ}(\lambda)} = {\begin{pmatrix}{\int{{S_{\lambda}\left( \lambda^{\prime} \right)}{\overset{\_}{x}\left( \lambda^{\prime} \right)}{\lambda^{\prime}}}} \\{\int{{S_{\lambda}\left( \lambda^{\prime} \right)}{\overset{\_}{y}\left( \lambda^{\prime} \right)}{\lambda^{\prime}}}} \\{\int{{S_{\lambda}\left( \lambda^{\prime} \right)}{\overset{\_}{z}\left( \lambda^{\prime} \right)}{\lambda^{\prime}}}}\end{pmatrix} = {\begin{pmatrix}{\overset{\_}{x}(\lambda)} \\{\overset{\_}{y}(\lambda)} \\{\overset{\_}{z}(\lambda)}\end{pmatrix} = {\overset{}{xyz}(\lambda)}}}} \\{{\overset{}{LMS}(\lambda)} = {\begin{pmatrix}{\int{{S_{\lambda}\left( \lambda^{\prime} \right)}{\overset{\_}{l}\left( \lambda^{\prime} \right)}{\lambda^{\prime}}}} \\{\int{{S_{\lambda}\left( \lambda^{\prime} \right)}{\overset{\_}{l}\left( \lambda^{\prime} \right)}{\lambda^{\prime}}}} \\{\int{{S_{\lambda}\left( \lambda^{\prime} \right)}{\overset{\_}{s}\left( \lambda^{\prime} \right)}{\lambda^{\prime}}}}\end{pmatrix} = {\begin{pmatrix}{\overset{\_}{l}(\lambda)} \\{\overset{\_}{m}(\lambda)} \\{\overset{\_}{s}(\lambda)}\end{pmatrix} = {\overset{}{lms}(\lambda)}}}}\end{matrix} & {{Eq}.\mspace{14mu} \left( {1\text{-}3} \right)}\end{matrix}$

When it is determined that a given set of R,G,B magnitudes oftristimulous colors {right arrow over (λ)}_(T) match a reference colorof wavelength λ, the values of XYZ for the matching colors must beequal, which implies the following:

$\begin{matrix}\begin{matrix}{{\overset{}{XYZ}(\lambda)} = {{{R\left( {\lambda,{\overset{->}{\lambda}}_{T}} \right)}{\overset{}{XYZ}\left( \lambda_{r} \right)}} +}} \\{{{G\left( {\lambda,{\overset{->}{\lambda}}_{T}} \right){\overset{}{XYZ}\left( \lambda_{g} \right)}} + {{B\left( {\lambda,{\overset{->}{\lambda}}_{T}} \right)}{\overset{}{XYZ}\left( \lambda_{b} \right)}}}} \\{= {{{R\left( {\lambda,{\overset{->}{\lambda}}_{T}} \right)}\begin{pmatrix}{\overset{\_}{x}\left( \lambda_{r} \right)} \\{\overset{\_}{y}\left( \lambda_{r} \right)} \\{\overset{\_}{z}\left( \lambda_{r} \right)}\end{pmatrix}} +}} \\{{{G\left( {\lambda,{\overset{->}{\lambda}}_{T}} \right)\begin{pmatrix}{\overset{\_}{x}\left( \lambda_{g} \right)} \\{\overset{\_}{y}\left( \lambda_{g} \right)} \\{\overset{\_}{z}\left( \lambda_{g} \right)}\end{pmatrix}} + {{B\left( {\lambda,{\overset{->}{\lambda}}_{T}} \right)}\begin{pmatrix}{\overset{\_}{x}\left( \lambda_{b} \right)} \\{\overset{\_}{y}\left( \lambda_{b} \right)} \\{\overset{\_}{z}\left( \lambda_{b} \right)}\end{pmatrix}}}} \\{= {{M_{\overset{\_}{xyz}}\left( {\overset{->}{\lambda}}_{T} \right)}{\overset{}{RGB}\left( {\lambda,{\overset{->}{\lambda}}_{T}} \right)}}} \\{= {\begin{pmatrix}{\overset{\_}{x}(\lambda)} \\{\overset{\_}{y}(\lambda)} \\{\overset{\_}{z}(\lambda)}\end{pmatrix}\mspace{14mu} {where}}}\end{matrix} & {{Eq}.\mspace{14mu} \left( {1\text{-}4} \right)} \\{{M_{\overset{\_}{xyz}}\left( {\overset{->}{\lambda}}_{T} \right)} = \begin{pmatrix}{{\overset{\_}{x}\left( \lambda_{r} \right)}{\overset{\_}{x}\left( \lambda_{g} \right)}{\overset{\_}{x}\left( \lambda_{b} \right)}} \\{{\overset{\_}{y}\left( \lambda_{r} \right)}{\overset{\_}{y}\left( \lambda_{g} \right)}{\overset{\_}{y}\left( \lambda_{b} \right)}} \\{{\overset{\_}{z}\left( \lambda_{r} \right)}{\overset{\_}{z}\left( \lambda_{g} \right)}{\overset{\_}{z}\left( \lambda_{b} \right)}}\end{pmatrix}} & {{Eq}.\mspace{14mu} \left( {1\text{-}5} \right)}\end{matrix}$

This implies that the predicted CMFs are given by:

$\begin{matrix}\begin{matrix}{{\overset{\rightarrow}{RGB}\left( {\lambda,{\overset{\rightarrow}{\lambda}}_{T}} \right)} = {\left( {M_{\overset{\_}{x}\overset{\_}{y}\overset{\_}{z}}\left( {\overset{\rightarrow}{\lambda}}_{T} \right)} \right)^{- 1}{\overset{\rightarrow}{XYZ}(\lambda)}}} \\{= {\left( {M_{\overset{\_}{x}\overset{\_}{y}\overset{\_}{z}}\left( {\overset{\rightarrow}{\lambda}}_{T} \right)} \right)^{- 1}{\overset{\rightarrow}{xyz}(\lambda)}}}\end{matrix} & {{Eq}.\mspace{11mu} \left( {1\text{-}6} \right)}\end{matrix}$

Thus, there is a well-defined relationship between a given choice ofx(λ),y(λ),z(λ) observer functions (for example 2 degree and 10 degree)and the CMFs that should be observed for a given set of tristimulouswavelengths {right arrow over (λ)}_(T).

The above relationship implies that all CMFs are linear transforms ofone another:

$\begin{matrix}\begin{matrix}{{\overset{\rightarrow}{RGB}\left( {\lambda,{\overset{\rightarrow}{\lambda}}_{T_{1}}} \right)} = {{M_{\overset{\_}{x}\overset{\_}{y}\overset{\_}{z}}^{- 1}\left( {\overset{\rightarrow}{\lambda}}_{T_{1}} \right)}{\overset{\rightarrow}{xyz}(\lambda)}}} \\{= {{M_{\overset{\_}{x}\overset{\_}{y}\overset{\_}{z}}^{1 -}\left( {\overset{\rightarrow}{\lambda}}_{T_{1}} \right)}{M_{\overset{\_}{x}\overset{\_}{y}\overset{\_}{z}}\left( {\overset{\rightarrow}{\lambda}}_{T_{2}} \right)}{M_{\overset{\_}{x}\overset{\_}{y}\overset{\_}{z}}^{- 1}\left( {\overset{\rightarrow}{\lambda}}_{T_{2}} \right)}{\overset{\rightarrow}{xyz}(\lambda)}}} \\{= {{M_{\overset{\_}{x}\overset{\_}{y}\overset{\_}{z}}^{- 1}\left( {\overset{\rightarrow}{\lambda}}_{T_{1}} \right)}{M_{\overset{\_}{x}\overset{\_}{y}\overset{\_}{z}}\left( {\overset{\rightarrow}{\lambda}}_{T_{2}} \right)}{\overset{\rightarrow}{RGB}\left( {\lambda,{\overset{\rightarrow}{\lambda}}_{T_{2}}} \right)}}}\end{matrix} & {{Eq}.\mspace{14mu} \left( {1\text{-}7} \right)}\end{matrix}$

This also implies that color matching is dependent only on the LMSsensitivities of the eye and is independent on the conversion matrixfrom CMFs to XYZ or LMS to XYZ:

$\begin{matrix}\begin{matrix}{{\overset{\rightarrow}{RGB}\left( {\lambda,{\overset{\rightarrow}{\lambda}}_{T}} \right)} = {\left( {M_{\overset{\_}{x}\overset{\_}{y}z}\left( {\overset{\rightarrow}{\lambda}}_{T} \right)} \right)^{- 1}{\overset{\rightarrow}{xyz}(\lambda)}}} \\{= {\left( {M_{\overset{\_}{x}\overset{\_}{y}\overset{\_}{z}}\left( {\overset{\rightarrow}{\lambda}}_{T} \right)} \right)^{- 1}M_{{XYZ}\rightarrow{LMS}}^{- 1}{\overset{\rightarrow}{lms}(\lambda)}}} \\{= {\left( {M_{{XYZ}\rightarrow{LMS}}^{- 1}{M_{\overset{\_}{l}\overset{\_}{m}\overset{\_}{s}}\left( {\overset{\rightarrow}{\lambda}}_{T} \right)}} \right)^{- 1}M_{{XYZ}\rightarrow{LMS}}^{- 1}{\overset{\rightarrow}{lms}(\lambda)}}} \\{= {\left( {M_{\overset{\_}{l}\overset{\_}{m}\overset{\_}{s}}\left( {\overset{\rightarrow}{\lambda}}_{T} \right)} \right)^{- 1}\left( M_{{XYZ}\rightarrow{LMS}}^{- 1} \right)^{- 1}}} \\{{M_{{XYZ}\rightarrow{LMS}}^{- 1}{\overset{\rightarrow}{lms}(\lambda)}}} \\{= {\left( {M_{\overset{\_}{l}\overset{\_}{m}\overset{\_}{s}}\left( {\overset{\rightarrow}{\lambda}}_{T} \right)} \right)^{- 1}M_{{XYZ}\rightarrow{LMS}}M_{{XYZ}\rightarrow{LMS}}^{- 1}{\overset{\rightarrow}{lms}(\lambda)}}} \\{= {\left( {M_{\overset{\_}{l}\overset{\_}{m}\overset{\_}{s}}\left( {\overset{\rightarrow}{\lambda}}_{T} \right)} \right)^{- 1}{\overset{\rightarrow}{lms}(\lambda)}}}\end{matrix} & {{Eq}.\mspace{14mu} \left( {1\text{-}8} \right)} \\{where} & \; \\{\begin{matrix}{{\overset{\rightarrow}{xyz}\left( \lambda_{r} \right)} = {M_{{XYZ}\rightarrow{LMS}}^{- 1}{\overset{\rightarrow}{lms}\left( \lambda_{r} \right)}}} \\{{\overset{\rightarrow}{xyz}\left( \lambda_{g} \right)} = {M_{{XYZ}\rightarrow{LMS}}^{- 1}{\overset{\rightarrow}{lms}\left( \lambda_{g} \right)}}} \\{{\overset{\rightarrow}{xyz}\left( \lambda_{b} \right)} = {M_{{XYZ}\rightarrow{LMS}}^{- 1}{\overset{\rightarrow}{lms}\left( \lambda_{b} \right)}}}\end{matrix}{{implies}\mspace{14mu} {that}}} & {{Eq}.\mspace{14mu} \left( {1\text{-}9} \right)} \\\begin{matrix}{{M_{\overset{\_}{x}\overset{\_}{y}\overset{\_}{z}}\left( {\overset{\rightarrow}{\lambda}}_{T} \right)} = {M_{{XYZ}\rightarrow{LMS}}^{- 1}\begin{pmatrix}{\overset{\_}{l}\left( \lambda_{r} \right)} & {\overset{\_}{l}\left( \lambda_{g} \right)} & {\overset{\_}{l}\left( \lambda_{b} \right)} \\{\overset{\_}{m}\left( \lambda_{r} \right)} & {\overset{\_}{m}\left( \lambda_{g} \right)} & {\overset{\_}{m}\left( \lambda_{b} \right)} \\{\overset{\_}{s}\left( \lambda_{r} \right)} & {\overset{\_}{s}\left( \lambda_{g} \right)} & {\overset{\_}{s}\left( \lambda_{b} \right)}\end{pmatrix}}} \\{= {M_{{XYZ}\rightarrow{LMS}}^{- 1}{M_{\overset{\_}{l}\overset{\_}{m}\overset{\_}{s}}\left( {\overset{\rightarrow}{\lambda}}_{T} \right)}}}\end{matrix} & {{Eq}.\mspace{14mu} \left( {1\text{-}10} \right)}\end{matrix}$

Since some reasonable estimates for the LMS sensitivities as a functionof wavelength are available, these functions can be characterized andrefined in a manner so as to obtain the best fit between prediction andresult for any existing CMF data sets. It can also be compared how wellany particular XYZ or LMS characterization fits the data using varioustristimulous {right arrow over (λ)}_(T) can also be performed.

The entire mathematical framework in Equations (1-1)-(1-10) above can becalculated by using the red, green, blue spectra of the CRT in lieu ofthe three Dirac delta functions for the monochromatic tristimulouswavelengths {right arrow over (λ)}_(T). One can easily show thatEquation 1-1 through 1-10 above now becomes:

$\begin{matrix}{{{\overset{\rightarrow}{RGB}\left( {\lambda,{{\overset{\rightarrow}{S}}_{T}\left( \lambda^{\prime} \right)}} \right)} = {\left( {M_{LMS}\left( {{\overset{\rightarrow}{S}}_{T}\left( \lambda^{\prime} \right)} \right)} \right)^{- 1}{\overset{\rightarrow}{lms}(\lambda)}}}{where}{{{\overset{\rightarrow}{S}}_{T}\left( \lambda^{\prime} \right)} = \begin{pmatrix}{S_{r}\left( \lambda^{\prime} \right)} \\{S_{g}\left( \lambda^{\prime} \right)} \\{S_{b}\left( \lambda^{\prime} \right)}\end{pmatrix}}} & {{Eq}.\mspace{14mu} \left( {1\text{-}11} \right)} \\{{M_{LMS}\left( {{\overset{\rightarrow}{S}}_{T}\left( \lambda^{\prime} \right)} \right)} = \begin{pmatrix}{L\left( {S_{r}\left( \lambda^{\prime} \right)} \right)} & {L\left( {S_{g}\left( \lambda^{\prime} \right)} \right)} & {L\left( {S_{b}\left( \lambda^{\prime} \right)} \right)} \\{M\left( {S_{r}\left( \lambda^{\prime} \right)} \right)} & {M\left( {S_{g}\left( \lambda^{\prime} \right)} \right)} & {M\left( {S_{b}\left( \lambda^{\prime} \right)} \right)} \\{S\left( {S_{r}\left( \lambda^{\prime} \right)} \right)} & {S\left( {S_{r}\left( \lambda^{\prime} \right)} \right)} & {S\left( {S_{r}\left( \lambda^{\prime} \right)} \right)}\end{pmatrix}} & {{Eq}.\mspace{14mu} \left( {1\text{-}12} \right)} \\{{{\overset{\rightarrow}{LMS}\left( {S_{r}\left( \lambda^{\prime} \right)} \right)} = {\int{{S_{r}\left( \lambda^{\prime} \right)}{\overset{\rightarrow}{lms}\left( \lambda^{\prime} \right)}{\lambda^{\prime}}}}}{{\overset{\rightarrow}{LMS}\left( {S_{g}\left( \lambda^{\prime} \right)} \right)} = {\int{{S_{g}\left( \lambda^{\prime} \right)}{\overset{\rightarrow}{lms}\left( \lambda^{\prime} \right)}{\lambda^{\prime}}}}}{{\overset{\rightarrow}{LMS}\left( {S_{b}\left( \lambda^{\prime} \right)} \right)} = {\int{{S_{b}\left( \lambda^{\prime} \right)}{\overset{\rightarrow}{lms}\left( \lambda^{\prime} \right)}{\lambda^{\prime}}}}}} & {{Eq}.\mspace{14mu} \left( {1\text{-}13} \right)}\end{matrix}$

Method for Improving LMS and XYZ with Existing Data

Having confirmed that LMS is all that is required to determine whethercolors match, next it is considered whether the LMS functions can beparameterized. If this is the case, one should be able to update the LMSfunctions in a progressive manner using data from 1931, 1955, and fromtoday as illustrative examples. One should be able to combine data fromthese experiments with color matching data involving complex spectra.

The most common experiments performed other than the color matchingexperiments described above are experiments that determine pairs ofvisual metamers. This typically involves having a fixed reference colorusing one medium, and comparative colors created using a differentmedium that can easily be adjusted or modified in order to determine avisual match, i.e. define a pair of visual metamers.

For example, if the spectra for an Apple Cinema display, an Eizo™ GC210display, and a GTI D50 viewer have been determined to match visually byadjusting the displays to match colors such as white paper in the GTIviewer, the spectra of the white shown on the display and the white ofthe paper in the viewer can be measured and converted to CIEXYZ andCIELAB. To the extant that the values of CIEXYZ and CIELAB aredifferent, one can assume inaccuracies in the human observer functions.Previous attempts to improve CIEXYZ and CIELAB to correct thesedifferences have not been successful. The present invention will addressthe issue via adjustments to the LMS cone responses and therefore to theXYZ functions derived from them while remaining consistent with datafrom the historical CMF experiments.

It is worthwhile to note that the differences observed between the 2 and10 degree observer are of similar magnitude to the variability in dataseems to indicate that there is still uncertainty in the definitions ofXYZ which a global optimization of LMS parameters could help to improve.

Studying the plots for LMS, it seems clear that to a first order theycan be considered asymmetrical Gaussians, which would be typicalbehavior of a quantum transition that is Doppler-broadened by rotationaland vibrational energy levels. The following very simplified model canbe to describe the LMS functions:

$\begin{matrix}{{f_{lms}\left( {\lambda,\alpha_{i},\lambda_{i},{\Delta\lambda}_{i\; 1},{\Delta\lambda}_{i\; 2},\gamma_{i\; 1},\gamma_{i\; 2},\delta_{i\; 1},\delta_{i\; 2},{\Delta\gamma}_{i\; 1},{\Delta\gamma}_{i\; 2}} \right)} = {{{{\alpha_{i}\left( {\delta_{i\; 1} + {\left( {1 - \delta_{i\; 1}} \right)^{- {({{{{\lambda - \lambda_{i}}}/2}{\Delta\lambda}_{i\; 1}})}^{({\gamma_{i\; 1} + {{\Delta\gamma}_{i\; 1}{{\lambda - \lambda_{i}}}}})}}}} \right)}\mspace{14mu} {for}\mspace{14mu} \lambda} < \lambda_{i}} = {{{\alpha_{i}\left( {\delta_{i\; 2} + \left( {1 - \delta_{i\; 2}} \right)} \right)}^{{- {({{{{\lambda - \lambda_{i}}}/2}\Delta \; \lambda_{i\; 2}})}^{({\gamma_{i\; 2} + {\Delta \; \gamma_{i\; 2}{{\lambda - \lambda_{i}}}}})}}\mspace{11mu}}{for}\mspace{14mu} \lambda} > \lambda_{i}}}} & {{Eq}.\mspace{14mu} \left( {2\text{-}1} \right)}\end{matrix}$

where i=0,1,2 for red (L), green (M), and blue (S). The parameter λ_(i)defines the wavelength of maximum sensitivity for the L,M, or Sfunction. Δλ_(1i) defines the width of the quasi-Gaussian on the sidewhere λ<λ_(i), Δλ_(2i) defines the width of the quasi-Gaussian forλ>λ_(i). The exponents γ_(1i) and γ_(2i) (which are nominally of value 2for a Gaussian distribution) in a similar fashion define the steepnessof the curve shape for a given Gaussian-like width for λ<λ_(i) andλ>λ_(i). The scaling parameter α_(i), defines the relative height of thesensitivity for LMS. Since equal energy spectra appear white to thehuman eye, it is generally assumed that the integral of L, M, and S areequal in value, which can be ensured by setting the appropriate value ofα_(i) for each.

The values of δ_(i1) and δ_(i2) (which are nominally 0) allow controlover the minimum value of the LMS functions—this is important since verysmall values of LMS can have a big impact on CIELAB due to the cube rootfunctions that define it. Finally, the correction parameters Δγ_(i1) andΔγ_(i2) (which are also nominally 0) allow a gradual increase ordecrease in the power law of the exponent to optimize the correlationbetween the parameterized LMS and XYZ vs. the officially accepted valuesof LMS and XYZ. Like the other parameters, they correspond to λ<λ_(i)and λ>λ_(i) respectively.

Since the focus has been on color matching between combinations ofnarrow band spectra of wavelength λ, the tristimulous values XYZ havebeen loosely characterized as functions of wavelength λ. Thetristimulous values XYZ will be distinguished from the human observerfunctions x(λ),y(λ),z(λ). The former is the integrated product of aspectral stimulant characterized by spectrum S(λ) and the correspondingobserver function x(λ),y(λ),z(λ). Similarly the tristimulous coneresponse values LMS are distinguished from the cone response functionsof the human eye l(λ),m(λ),s(λ):

X=∫S(λ)x(λ)dλ

Y=∫S(λ)y(λ)dλ

Z=∫S(λ)z(λ)dλ

L=∫S(λ)l(λ)dλ

M=∫S(λ)m(λ)dλ

S=∫S(λ)s(λ)dλ  Eq. (2-2)

Thus, a parameterized x(λ),y(λ),z(λ) can be created from a parameterizedl(λ),m(λ),s(λ) as follows in Eq. 2-3 below:

$\begin{matrix}{{{\overset{\rightarrow}{XYZ}(\lambda)} = {M_{{LMS}\rightarrow{XYZ}}{\overset{\rightarrow}{LMS}(\lambda)}}}{{\overset{\rightarrow}{xyz}(\lambda)} = {\begin{pmatrix}{\overset{\_}{x}(\lambda)} \\{\overset{\_}{y}(\lambda)} \\{\overset{\_}{z}(\lambda)}\end{pmatrix} = {{M_{{LMS}\rightarrow{XYZ}}{\overset{\rightarrow}{lms}(\lambda)}} = {M_{{LMS}\rightarrow{XYZ}}\begin{pmatrix}{\overset{\_}{l}(\lambda)} \\{\overset{\_}{m}(\lambda)} \\{\overset{\_}{s}(\lambda)}\end{pmatrix}}}}}{{{\overset{\rightarrow}{xyz}}_{E}(\lambda)} = {M_{{LMS}\rightarrow{XYZ}}\begin{pmatrix}{f_{lms}\left( {\lambda,\alpha_{l},\lambda_{l},{\Delta \; \lambda_{l\; 1}},{\Delta \; \lambda_{l\; 2}},\gamma_{l\; 1},\gamma_{l\; 2},\delta_{l\; 1},\delta_{l\; 2},{\Delta \; \gamma_{l\; 1}},{\Delta \; \gamma_{l\; 2}}} \right)} \\{f_{lms}\left( {\lambda,\alpha_{m},\lambda_{m},{\Delta \; \lambda_{m\; 1}},{\Delta \; \lambda_{m\; 2}},\gamma_{m\; 1},\gamma_{m\; 2},\delta_{m\; 1},\delta_{m\; 2},{\Delta\gamma}_{m\; 1},{\Delta\gamma}_{m\; 2}} \right)} \\{f_{lms}\left( {\lambda,\alpha_{s},\lambda_{s},{\Delta\lambda}_{s\; 1},{\Delta \; \lambda_{s\; 2}},\gamma_{s\; 1},\gamma_{s\; 2},\delta_{s\; 1},\delta_{s\; 2},{\Delta\gamma}_{s\; 1},{\Delta\gamma}_{s\; 2}} \right)}\end{pmatrix}}}} & \left( {2\text{-}3} \right)\end{matrix}$

where

M _(LMS->XYZ) =M _(XYZ->LMS) ⁻¹   (Eq. 2-3a)

The {right arrow over (xyz)}_(E) is used to denote the “EdgeXYZ” humanobserver functions x(λ),y(λ),z(λ) in accordance with one embodiment ofthe invention and {right arrow over (XYZ)}_(E) is used to denote theintegrated tristimulous values XYZ calculated from the color stimulusand {right arrow over (xyz)}_(E). In the equation above, the dependenceof {right arrow over (xyz)}_(E) on the EdgeXYZ parameters is implicit.In later expressions it will be explicitly indicated that {right arrowover (xyz)}_(E) is a function of the EdgeXYZ parameters.

The following values in Table 1 were determined via manual adjustment ofthe above parameters, using nominal values of 0.0 for δ's and Δγ's:

TABLE 1 Value (based on 2 degree CMFs of G&W) L M S Lambda0 577 547 449MaxTristim 0.95 1.1 1.77 Width1 32.6 24.0 16.0 Width2 29.3 30.0 18.4Exp1 2.6 1.7 2.8 Exp2 2.25 1.94 1.8The resulting plots for LMS are shown in FIG. 1. FIGS. 2 and 3 highlightthe color matching functions predicted versus data at fill scale for Rand magnified×25 for G and B. Finally, FIG. 4 shows actual XYZ observerfunctions vs. the results of calculating LMS and converting to XYZ viathe inverse of the Hunt-Pointer-Estivez matrix.

The simple parameterization of LMS above gives surprisingly goodresults. A least squares fit was performed in order to optimize theparameters above using the spectral cmf data of Guild and Wright. Thecost function to be minimized was a combination of ΔE difference betweenCIEXYZ and EdgeXYZ for monochromatic light for D50 illumination for awhite reflector and for 32 monochromatic light source spectra of power ⅓of the corresponding white reflector ranging from 380 nm to 730 nm andthe difference between the spectral response of CIEXYZ and EdgeXYZ witha weighting factor of 100 in order to correspond roughly to the range ofCIELAB. The average and max ΔE results were calculated for the aboveafter performing the LSF, and separate test data was used to confirmfurther the validity of the EdgeXYZ vs. CIEXYZ functions. The test datacomprised of 262 reflective spectral measurements of Matchprint Digitalsamples, including all permutations of 0%, 40%, 70%, and 100% tints forCMYK.

The following parameters shown in Table 2 below give an average error of1.2 ΔE and a maximum error of 6.0 ΔE between CIEXYZ for the 2 degreeobserver and the EdgeXYZ for D50 white and for the extreme case of themonochromatic light stimuli (noting that typical values of chroma were100-250), and average and max errors of 0.2 ΔE and 0.6 ΔE respectivelyfor the 262 test Matchprint colors:

TABLE 2 Calculated Parameters for EdgeXYZ XYZParamName LMS_L LMS_M LMS_Sm_LMS_alpha = 0.946 1.088 1.777 m_LMS_lambda = 575.839 546.015 451.285m_LMS_gamma1 = 2.806 1.903 4.067 m_LMS_gamma2 = 2.507 2.150 1.840m_LMS_DeltaLambda1 = 32.786 24.490 15.920 m_LMS_DeltaLambda2 = 30.02330.563 17.000 m_LMS_delta1 = −0.006 −0.003 −0.032 m_LMS_delta2 = −0.003−0.002 −0.003 m_LMS_DeltaGamma1 = −0.005 −0.011 −0.009 m_LMS_DeltaGamma2= −0.001 −0.002 −0.001This resulted in the plots for CIEXYZ vs. EdgeXYZ as shown in FIG. 5. Ina similar fashion, the EdgeXYZ parameterization was performed on theCIEXYZ 10-degree observer functions, resulting in an average and maxerror between the model and the standard observer of 0.98 ΔE and 5.1 ΔEfor D50 white and for the monochromatic colors of the visible spectrumas shown in Table 3, results are shown in FIG. 6.

TABLE 3 Calculated Parameters for EdgeXYZ XYZParamName LMS_L LMS_M LMS_Sm_LMS_alpha = 0.991 1.073 2.000 m_LMS_lambda = 572.880 543.272 450.040m_LMS_gamma1 = 2.315 1.652 2.571 m_LMS_gamma2 = 2.844 2.075 1.839m_LMS_DeltaLambda1 = 34.080 28.833 17.021 m_LMS_DeltaLambda2 = 31.06933.034 15.813 m_LMS_delta1 = −0.003 0.001 0.004 m_LMS_delta1 = −0.0050.000 −0.004 m_LMS_DeltaGamma1 = 0.000 −0.002 −0.002 m_LMS_DeltaGamma2 =0.000 0.000 −0.002

The fact that the EdgeXYZ model can simulate either the 2 or 10 degreeobserver to an average of 1ΔE for the extreme case of monochromaticcolors is a very good indicator that the model should be satisfactoryfor optimizing the existing CIEXYZ observer based on all available cmfdata. Since 1ΔE is a good confirmation of a good model, one should alsoask how consistent are the two existing standards (2 and 10 degree) toone another. This comparison can be performed either by calculating XYZfor each observer for D50 and for each monochromatic color of wavelengthλ in order to calculate L*a*b* and therefore ΔE, or by calculatingvalues of RGB for a particular set of tristimulous colors {right arrowover (λ)}_(T) in order to match each monochromatic color of wavelength λfor one observer and asking how large is the calculated ΔE between thesetwo “matching” colors according to the other observer (note that anynegative values of R,G, or B can be added to the monochromatic stimuliin order to ensure positive values of calculated XYZ for the twocolors).

Thus, when the same quality criteria is applied for comparing theconsistency of the 2 and 10 degree standards to one another, it is foundthat pairs of saturated colors (such as used to obtain cmfs) that arepredicted to be a match by the 2 degree observer generally will not bepredicted to match by the 10 degree observer, and vice versa. In fact,the two observers disagree by an average and maximum error of 11 ΔE and76 ΔE, respectively, for saturated colors. If both models are used tooptimize the EdgeXYZ model, the model will now have an average and maxerror of 6 ΔE and 40 ΔE for each of the two observers for themonochromatic colors of the spectrum.

The validity of the above assessment can be confirmed by simple visualcomparison of the 2 and 10-degree observers as shown in FIG. 7 where theaccurate EdgeXYZ model for 10 degree has been used to compare with the 2degree standard with a similar white normalization. The plot showssignificant differences, particularly in the vicinity of wavelength 480nm where we can clearly see values of Y=0.14 and Y=0.24 for the 2 degreeand 10 degree observers, respectively. This seemingly small error of 10%is retained after the cube root functions contained in the expressionsfor CIELAB are applied. This results in a 10ΔE difference in L* (due tothe factor of 100 in the equations for L*) and a 50 ΔE difference in a*(due to the factor of 500 in the equations for a*). It should be notedthat the 10 degree cmfs obtained by Stiles and Burch were actually arepeat of 2 degree cmfs they obtained. The differences between thestandard 2 and 10 degree observers was never truly reconciled, therebyresulting in 2 conflicting standards.

If one observes a split circle containing two adjacent matching colorswith a size equivalent to 2 degrees, that pair of matching colors willnot differ in appearance by 50 ΔE merely by increasing its size to theequivalent of 10 degrees.

Since there is quite a large gap between the 2 and 10 degree observerfunctions, it may well be that one or the other is a betterrepresentation of human color vision. It is proposed therefore thatimproved CIEXYZ functions be determined by optimization of the EdgeXYZparameters demonstrated in the above examples. This optimization can usethe 2 degree observer functions as the baseline or the 10 degreefunctions. The optimization includes a diverse sampling of metamericpairs of colors with significantly different spectral powerdistributions (SPDs), neutral white in particular since (as pointed outby Fairchild) the eye is very sensitive to gray balance. Metamers ormetameric pairs are stimuli that are spectrally different but visuallyidentical to the human eye.

The optimization above will now be farther clarified and modified toinclude pairs of metameric matches with significantly different SPDs inorder to determine optimized human observer functions. Starting with thehuman observer functions, the functions of λ and the EdgeXYZcharacterization parameters are as follows:

$\begin{matrix}{{{{\overset{\rightarrow}{xyz}}_{E}\left( {\lambda,\overset{\rightarrow}{\alpha},\overset{\rightarrow}{\lambda},{\overset{\rightarrow}{\Delta \; \lambda}}_{1},{\overset{\rightarrow}{\Delta \; \lambda}}_{2},{\overset{\rightarrow}{\gamma}}_{1},{\overset{\rightarrow}{\gamma}}_{2},{\overset{\rightarrow}{\delta}}_{1},{\overset{\rightarrow}{\delta}}_{2},{\overset{\rightarrow}{\Delta \; \gamma}}_{1},{\overset{\rightarrow}{\Delta \; \gamma}}_{2}} \right)} = {M_{{LMS}\rightarrow{XYZ}}{{\overset{\rightarrow}{lms}}_{E}\left( {\lambda,\overset{\rightarrow}{\alpha},\overset{\rightarrow}{\lambda},{\overset{\rightarrow}{\Delta \; \lambda}}_{1},{\overset{\rightarrow}{\Delta \; \lambda}}_{2},{\overset{\rightarrow}{\gamma}}_{1},{\overset{\rightarrow}{\gamma}}_{2},{\overset{\rightarrow}{\delta}}_{1},{\overset{\rightarrow}{\delta}}_{2},{\overset{\rightarrow}{\Delta\gamma}}_{1},{\overset{\rightarrow}{\Delta\gamma}}_{2}} \right)}}}{where}} & {{Eq}.\mspace{14mu} \left( {3\text{-}1} \right)} \\\begin{matrix}{{\overset{\rightarrow}{\alpha} = \begin{pmatrix}\alpha_{L} \\\alpha_{M} \\\alpha_{S}\end{pmatrix}},{\overset{\rightarrow}{\lambda} = \begin{pmatrix}\lambda_{L} \\\lambda_{M} \\\lambda_{S}\end{pmatrix}},{\overset{\rightarrow}{{\Delta\lambda}_{1}} = \begin{pmatrix}{\Delta\lambda}_{1L} \\{\Delta\lambda}_{1M} \\{\Delta\lambda}_{1S}\end{pmatrix}},{\overset{\rightarrow}{{\Delta\lambda}_{2}} = \begin{pmatrix}{\Delta\lambda}_{2L} \\{\Delta\lambda}_{2M} \\{\Delta\lambda}_{2S}\end{pmatrix}},} \\{{{\overset{\rightarrow}{\gamma}}_{1} = \begin{pmatrix}\gamma_{1L} \\\gamma_{1M} \\\gamma_{1S}\end{pmatrix}},{{\overset{\rightarrow}{\gamma}}_{2} = \begin{pmatrix}\gamma_{2L} \\\gamma_{2M} \\\gamma_{2S}\end{pmatrix}},{{\overset{\rightarrow}{\delta}}_{1} = \begin{pmatrix}\delta_{1L} \\\delta_{1M} \\\delta_{1S}\end{pmatrix}},{{\overset{\rightarrow}{\delta}}_{2} = \begin{pmatrix}\delta_{2L} \\\delta_{2M} \\\delta_{2S}\end{pmatrix}},} \\{{\overset{\rightarrow}{{\Delta\gamma}_{1}} = \begin{pmatrix}{\Delta\gamma}_{1L} \\{\Delta\gamma}_{1M} \\{\Delta\gamma}_{1S}\end{pmatrix}},{\overset{\rightarrow}{{\Delta\gamma}_{2}} = \begin{pmatrix}{\Delta\gamma}_{2L} \\{\Delta\gamma}_{2M} \\{\Delta\gamma}_{2S}\end{pmatrix}}}\end{matrix} & {{Eq}.\mspace{14mu} \left( {3\text{-}2} \right)}\end{matrix}$

The cost function to be minimized is the sum of two summations. Thefirst summation combines the squared errors between EdgeXYZ humanobserver functions x(λ),y(λ),z(λ) and CIEXYZ human observer functionsx(λ),y (λ),z(λ) (either 2 degree, 10 degree, or both). The firstsummation extends over N_(λ) wavelengths across the visible spectrum.The second summation combines the squared ΔE errors between calculatedpairs of visually matching spectra S_(ij)(λ), where i=1 to N_(s), thenumber of sample pairs and j=1, 2 denotes each of the two samples ineach pair:

$\begin{matrix}{{\left. {{{Err}\left( {\overset{\rightarrow}{\alpha},\overset{\rightarrow}{\lambda},{\overset{\rightarrow}{\Delta\lambda}}_{1},{\overset{\rightarrow}{\Delta\lambda}}_{2},{\overset{\rightarrow}{\gamma}}_{1},{\overset{\rightarrow}{\gamma}}_{2},{\overset{\rightarrow}{\delta}}_{1},{\overset{\rightarrow}{\delta}}_{2},{\overset{\rightarrow}{\Delta\gamma}}_{1},{\overset{\rightarrow}{\Delta\gamma}}_{2}} \right)} = {{100.0{\sum\limits_{i = 1}^{i = N_{\lambda}}\left\lbrack {{\overset{\rightarrow}{xyz}}_{E}\left( {\lambda_{i},\overset{\rightarrow}{\alpha},\overset{\rightarrow}{\lambda},{\overset{\rightarrow}{\Delta\lambda}}_{1},{\overset{\rightarrow}{\Delta\lambda}}_{2}, {\overset{\rightarrow}{\gamma}}_{1}, {\overset{\rightarrow}{\gamma}}_{2}, {\overset{\rightarrow}{\delta}}_{1}, {\overset{\rightarrow}{\delta}}_{2},{\overset{\rightarrow}{\Delta\gamma}}_{1},{\overset{\rightarrow}{\Delta\gamma}}_{2}} \right)} \right)}} - {\overset{\rightarrow}{xyz}\left( \lambda_{i} \right)}}} \right\rbrack^{2} + {\sum\limits_{i = 1}^{i = N_{s}}\left\lbrack {{\overset{\rightarrow}{Lab}\left( {{\overset{\rightarrow}{XYZ}}_{E}\left( {{S_{i\; 1}(\lambda)},\overset{\rightarrow}{\alpha},\overset{\rightarrow}{\lambda},{\overset{\rightarrow}{\Delta\lambda}}_{1},{\overset{\rightarrow}{\Delta\lambda}}_{2}, {\overset{\rightarrow}{\gamma}}_{1}, {\overset{\rightarrow}{\gamma}}_{2}, {\overset{\rightarrow}{\delta}}_{1}, {\overset{\rightarrow}{\delta}}_{2},{\overset{\rightarrow}{\Delta\gamma}}_{1},{\overset{\rightarrow}{\Delta\gamma}}_{2}} \right)} \right)} - {\overset{\rightarrow}{{Lab}\;}\left( {{\overset{\rightarrow}{XYZ}}_{E}\left( {{S_{i\; 2}(\lambda)},\overset{\rightarrow}{\alpha},\overset{\rightarrow}{\lambda},{\overset{\rightarrow}{\Delta\lambda}}_{1},{\overset{\rightarrow}{\Delta\lambda}}_{2}, {\overset{\rightarrow}{\gamma}}_{1}, {\overset{\rightarrow}{\gamma}}_{2}, {\overset{\rightarrow}{\delta}}_{1}, {\overset{\rightarrow}{\delta}}_{2},{\overset{\rightarrow}{\Delta\gamma}}_{1},{\overset{\rightarrow}{\Delta\gamma}}_{2}} \right)} \right)}} \right\rbrack^{2}}}{where}} & {{Eq}.\mspace{14mu} \left( {3\text{-}3} \right)} \\{{{\overset{\rightarrow}{XYZ}}_{E}\left( {{S_{ij}(\lambda)},\overset{\rightarrow}{\alpha},\overset{\rightarrow}{\lambda},{\overset{\rightarrow}{\Delta\lambda}}_{1},{\overset{\rightarrow}{\Delta \; \lambda}}_{2},{\overset{\rightarrow}{\gamma}}_{1},{\overset{\rightarrow}{\gamma}}_{2},{\overset{\rightarrow}{\delta}}_{1},{\overset{\rightarrow}{\delta}}_{2},{\overset{\rightarrow}{\Delta\gamma}}_{1},{\overset{\rightarrow}{\Delta\gamma}}_{2}} \right)} = {\int{{S_{ij}(\lambda)}{{xyz}_{E}\left( {\lambda,\overset{\rightarrow}{\alpha},\overset{\rightarrow}{\lambda},{\overset{\rightarrow}{\Delta\lambda}}_{1},{\overset{\rightarrow}{\Delta\lambda}}_{2},{\overset{\rightarrow}{\gamma}}_{1},{\overset{\rightarrow}{\gamma}}_{2},{\overset{\rightarrow}{\delta}}_{1},{\overset{\rightarrow}{\delta}}_{2},{\overset{\rightarrow}{\Delta\gamma}}_{1},{\overset{\rightarrow}{\Delta\gamma}}_{2}} \right)}{\lambda}}}} & {{Eq}.\mspace{14mu} \left( {3\text{-}4} \right)}\end{matrix}$

Note that the XYZ values of D50 white must be calculated using EdgeXYZand included in the calculation of L*a*b* from XYZ.

In order to include the historical cmf data that was used to createeither the 2 degree or the 10 degree observer functions into the aboveoptimization, matching spectral pairs can be constructed usingtristimulous intensities R, G, B with spectra in the form of Dirac deltafunctions and a reference color of wavelength λ (also in the form of aDirac delta function) which have the same values of XYZ using standardobserver functions.

Alternatively, one can calculate XYZ_(i1) and XYZ_(i2) for themonochromatic light using EdgeXYZ and the CIEXYZ observer functions.There are several mathematically equivalent methods in order to includethe historical cmfs: one key is to add pairs of spectra for an adequatesampling of λ that are “matching pairs” as defined by a particular setof CIEXYZ observer functions.

The above cost function is minimized using such methods as Powell'smethod by optimization of the EdgeXYZ parameters. This was the methodused to construct the examples above for the 2 and 10 degree observers.All that remains is to add more pairs of visually matching spectra inorder to minimize the cost function thereby resulting in improved XYZhuman observer functions.

A significant research effort was performed on this topic and publishedby Thornton, resulting in a multi-part series of articles in ColorResearch and Applications. The heart of his work, which evolved from hisexperience in the fluorescent lighting industry, was to create fivewhite light sources with extremely different SPDs that were assessed byseveral observers as being good visual metameric matches to one another.The fact that the values of XYZ and CIELAB that resulted from the SPDsof these five light sources were significantly different led toThornton's conclusion that the human observer functions requiredsignificant improvement. The numerical data characterizing the SPD ofthese sources was never made public. However, the SPD of one pair ofwhites was communicated to Mark Shaw of RIT. The SPDs of the five whitelight sources were presented in Thornton's publications in the form ofgraphs. By digitizing these graphs, the original data was extracted. Bycomparing two of the extracted data sets with the pair of SPDscommunicated to Shaw, it was confirmed that the data extracted fromThornton's graphs was reasonably accurate and could be used forvalidating and optimizing CIEXYZ.

It was confirmed that the 2 degree observer calculates significant ΔEdifferences between these five matching white light sources and theiraverage calculated L*a*b*. The following table calculates L*a*b* and XYZreference values based on the 2 degree observer for the five lightsources as shown in Table 4 below.

TABLE 4 L* ref a* ref b* ref X ref Y ref Z ref 55.68 −27.26 15.56 17.2523.60 13.00 53.74 −19.42 11.01 17.15 21.73 13.44 55.09 −11.98 7.18 19.6923.01 15.84 53.61 −5.76 5.63 19.65 21.60 15.43 54.44 −12.15 6.34 19.1022.39 15.73Although the values of L* are similar, there are clearly largedisagreements between the calculated values of a* and b*. Table 5 wascalculated using EdgeXYZ optimized for the 2 degree observer. Alsocalculated are the corresponding ΔE errors between each white lightsource and the average of all the sources:

TABLE 5 L* Edge a* Edge b* Edge X Edge Y Edge Z Edge Delta E 56.00−28.61 15.04 17.25 23.91 13.39 14.17 54.04 −20.84 10.46 17.13 22.0113.84 5.16 55.25 −12.17 6.75 19.79 23.17 16.13 4.37 53.32 −6.24 5.2019.31 21.34 15.40 10.46 54.25 −12.23 6.07 18.92 22.21 15.70 4.63 Delta E= 7.76 Delta E = 14.17

Optimizing the EdgeXYZ parameters to minimize this large ΔE error can beperformed as described in Equation (3-3) above. Since the original cmfdata was obtained using saturated colors and had significant variabilityas shown in FIG. 8-1 in the next section, and since Thornton's set of 5metameric colors were all matching whites as confirmed by 8 observers(white balance being very sensitive to the eye), a weighting factor of 3was chosen to give preference to minimizing the ΔE error for thematching whites vs. the ΔE error calculated from the cmf data of the 2degree observer. This weighting factor was achieved by dividing each ofthe two summations in the cost function indicated in Equation (3-3) byN_(λ) and N_(s) respectively in order to define the average sum squarederror for each of the two summations, then multiplying the lattersummation by 9.0 (i.e. 3² since a weighting factor of 3 in ΔE implies aweighting factor of 9 in the square of ΔE).

The resulting minimization of the cost function gave the followingEdgeXYZ parameters shown in Table 6.

TABLE 6 Calculated Parameters for EdgeXYZ XYZParamName LMS_L LMS_M LMS_Sm_LMS_alpha = 0.9865 1.1358 1.6802 m_LMS_lambda = 581.2820 548.3070451.7720 m_LMS_gamma1 = 3.2521 1.7375 4.4494 m_LMS_gamma2 = 2.42812.1655 1.6806 m_LMS_DeltaLambda1 = 32.8056 22.8717 16.5772m_LMS_DeltaLambda2 = 28.5968 30.4839 17.7028 m_LMS_delta1 = −0.0087−0.0047 −0.0351 m_LMS_delta1 = −0.0022 −0.0014 0.0002 m_LMS_DeltaGamma1= −0.0052 −0.0350 −0.0035 m_LMS_DeltaGamma2 = 0.0020 0.0013 0.0017The resulting minimization provided the plot of EdgeXYZ versus the 2degree standard observer as shown in FIG. 8. The improvement to thecalculated ΔE between the five matching whites was significant as shownin Table 7.

TABLE 7 L* Edge a* Edge b* Edge X Edge Y Edge Z Edge Delta E 54.05−13.54 11.52 18.51 22.03 13.45 3.01 53.02 −13.25 9.18 17.71 21.06 13.661.89 55.42 −12.29 8.40 19.92 23.34 15.58 1.66 53.84 −10.24 8.56 18.9621.82 14.43 1.58 54.06 −9.10 8.11 19.38 22.03 14.75 2.79 ave Delta E =2.19 max Delta E = 3.01The conclusion for the above analysis is that the CIEXYZ 2-degreeobserver is quite reasonable qualitatively. The observer data itself isvalid but it merely contained ± uncertainty that was never fullyconfronted.

The above analysis clearly illustrates that if the 2-degree observer isvaried within the known uncertainty of the data variability, sets ofmatching whites such as the light sources can be dramatically improvedin terms of the predicted ΔE match between them. Rather than attemptingto do this by varying all 32 discrete wavelengths or by varyingfunctions that have no physical basis for the CIE observer functions,the obvious optimization is to vary their key fundamental attributessuch as: the max wavelength, the width and shape of each side of thequasi-Gaussian peaks of the l(λ),m(λ),s(λ) sensitivity curves, and ifnecessary other slight empirical corrections as illustrated above.

Error minimization was performed on a combination of Thornton data aboveand corresponding Alvin data from Rochester Institute of Technology(RIT) that compared neutral gray print and transparency to thecorresponding RGB color on a CRT. Table 8 indicates the errors betweenpairs of metamers calculated with CIEXYZ using the EdgeXYZ model 2degree observer:

TABLE 8 L* Edge a* Edge b* Edge X Edge Y Edge Z Edge Delta E 56.00−28.61 15.04 17.25 23.91 13.39 0.00 54.04 −20.84 10.46 17.13 22.01 13.849.23 56.00 −28.61 15.04 17.25 23.91 13.39 0.00 55.25 −12.17 6.75 19.7923.17 16.13 18.43 56.00 −28.61 15.04 17.25 23.91 13.39 0.00 54.25 −12.236.07 18.92 22.21 15.70 18.76 56.00 −28.61 15.04 17.25 23.91 13.39 0.0053.32 −6.24 5.20 19.31 21.34 15.40 24.58 100.02 −15.24 −6.35 87.92100.06 90.65 0.00 99.99 −18.47 −14.62 86.11 99.98 101.92 8.88 100.04−9.50 7.88 91.13 100.10 73.20 0.00 99.92 −11.94 3.95 89.48 99.78 77.524.62 ave Delta E = 14.08 max Delta E = 24.58

In Table 9, the results of calculating the metameric pairs afteroptimizing the EdgeXYZ parameters are shown.

TABLE 9 L* Edge a* Edge b* Edge X Edge Y Edge Z Edge Delta E 54.15−10.34 10.66 19.21 22.11 13.83 0.00 53.17 −11.25 10.21 18.22 21.21 13.381.41 54.15 −10.34 10.66 19.21 22.11 13.83 0.00 54.65 −10.05 9.99 19.6922.59 14.41 0.88 54.15 −10.34 10.66 19.21 22.11 13.83 0.00 53.52 −10.649.29 18.63 21.53 13.94 1.54 54.15 −10.34 10.66 19.21 22.11 13.83 0.0055.18 −8.53 11.75 20.47 23.10 14.09 2.35 99.53 −9.01 −4.60 90.17 98.7987.27 0.00 99.82 −9.98 −9.72 90.32 99.53 94.69 5.22 99.53 −0.24 8.8595.12 98.80 71.11 0.00 100.34 −2.44 9.06 95.86 100.88 72.45 2.35 aveDelta E = 2.29 max Delta E = 5.22Table 10 shows the parameters that achieved the results above.

TABLE 10 Calculated Parameters for EdgeXYZ XYZParamName LMS_L LMS_MLMS_S m_LMS_alpha = 0.940 1.140 1.806 m_LMS_lambda = 579.036 542.416442.246 m_LMS_gamma1 = 3.196 1.878 2.400 m_LMS_gamma2 = 3.962 1.7681.179 m_LMS_DeltaLambda1 = 31.569 18.619 11.883 m_LMS_DeltaLambda2 =30.837 34.646 21.228 m_LMS_delta1 = 0.018 −0.006 −0.009 m_LMS_delta1 =−0.012 0.005 0.007 m_LMS_DeltaGamma1 = 0.017 −0.006 0.010m_LMS_DeltaGamma2 = 0.002 0.002 0.002

Table 6-9

In FIG. 9, the plots of EdgeXYZ vs. CIEXYZ (2 degree observer) arehighlighted.

It is interesting to note that the primary impact of the Alvin datatogether with the Thornton data is to sharpen the characterization ofthe EdgeXYZ_Z observer function. This rather sharp behavior may besmoothed by attempting to split the single quasi-Gaussian s(λ) blue coneresponse into two overlapping quasi-Gaussians in one embodiment. Thiswould require fitting an extra set of parameters for peak wavelength,left/right width, left/right exponential power, etc.

In view of the above results, and in view of the characterization ofPokorney et al. who observed structure in the s(λ) blue cone response, asecond quasi-Gaussian was added to characterize s(λ). This secondGaussian was constrained to be higher in value of peak λ and to vary inmagnitude between 0.0 and 0.9 of the magnitude of the primaryquasi-Gaussian defining s(λ). In order to minimize the number ofadjustable parameters, the same values of width, gamma, etc. were usedfor both curves defining s(λ). Only the amplitude and peak wavelength ofthis second curve was permitted to vary independently.

The resulting characterization of s(λ) had approximately the sameimprovement to the data set of Thornton and Alvin as observed above.However, the maximum error between this modified set of curves vs. thehistorical 1931 cmf data (i.e. errors for maximum saturatedmonochromatic colors) was reduced from 48 ΔE to 25 ΔE.

Table 11 below indicates the parameters for L and M, and for the twoquasi-Gaussian peaks S1 and S2 which were combined to form S(λ):

TABLE 11 Calculated Parameters for EdgeXYZ XYZParamName LMS_L LMS_MLMS_S1 LMS_S2 m_LMS_alpha = 0.929 1.119 1.379 0.914 m_LMS_lambda =580.331 545.162 436.343 462.305 m_LMS_gamma1 = 3.622 1.840 1.206 1.192m_LMS_gamma2 = 3.216 1.956 1.206 1.192 m_LMS_DeltaLambda1 = 32.81520.435 7.309 16.474 m_LMS_DeltaLambda2 = 29.825 32.730 7.309 16.474m_LMS_delta1 = −0.007 −0.005 0.010 0.007 m_LMS_delta1 = −0.007 0.0020.010 0.007 m_LMS_DeltaGamma1 = 0.007 −0.011 −0.005 0.008m_LMS_DeltaGamma2 = 0.003 0.002 −0.005 0.008

The plots for z(λ) (and hence for s(λ)) shown in FIG. 10. The above ismerely an example of how x(λ),y(λ),z(λ) can be parameterized.Illustrative examples of mathematical modeling to achieve even greateraccuracy relative to existing standards in accordance with otherembodiments are as follows:

-   -   1) Adding more parameters to characterize the LMS cone responses        using, for example, splines with a limited number of knot        points; and    -   2) Use existing CIEXYZ observer functions x(λ),y(λ),z(λ) (either        2 or 10 degree) as the baseline function and create a        Δx(λ),Δy(λ),Δz(λ) function using the parameterized EdgeXYZ        functions x(λ),y(λ),z(λ) above.

These additional methods may not be required since the EdgeXYZparameterization described above appears to be an adequate model forboth 2 and 10-degree observer functions. It also appears to be adequaterelative to the current noise and inconsistencies of the historical cmfexperiments. However, if desired, method #2 above has the advantage ofensuring that the exact original properties of CIEXYZ are preserved forΔx(λ),Δy(λ),Δz(λ)=0.0, even slight imperfections in shapes of the LMSand Δx(λ),Δy(λ),Δz(λ) curves due to errors in the original humanobserver data.

The following example of this approach will use a simplified EdgeXYZthat does not adjust the minimum offset or the exponents used in thequasi-Gaussian functions for lms. The mathematical expression for theparameterized Δx(λ),Δy(λ),Δz(λ) is as follows:

{right arrow over (xyz)} _(E)(λ,{right arrow over (α)},{right arrow over(λ)},{right arrow over (Δλ₁)},{right arrow over (Δλ₂)})={right arrowover (xyz)} _(CIE)(λ)+Δ{right arrow over (xyz)}(λ,{right arrow over(α)},{right arrow over (λ)},{right arrow over (Δλ₁)},{right arrow over(Δλ₂)})   Eq. (3-5)

where

Δ{right arrow over (xyz)} (λ,{right arrow over (α)},{right arrow over(λ)},{right arrow over (Δλ₁)},{right arrow over (Δλ₂)})={right arrowover (xyz)} _(E)(λ,{right arrow over (α)},{right arrow over (λ)},{rightarrow over (Δλ₁)},{right arrow over (Δλ₂)})−{right arrow over (xyz)}_(E)(λ,{right arrow over (α₀)},{right arrow over (λ₀)},{right arrow over(Δλ₀₁)},{right arrow over (Δλ₀₂)})

{right arrow over (xyz)} _(E)(λ,{right arrow over (α)},{right arrow over(λ)},{right arrow over (Δλ₁)},{right arrow over (Δλ₂)})=M _(LMS->XYZ){right arrow over (lms)} _(E)(λ,{right arrow over (α)},{right arrow over(λ)},{right arrow over (Δλ₁)},{right arrow over (Δλ₂)})

{right arrow over (xyz)} _(E)(λ,{right arrow over (α₀)},{right arrowover (λ)}₀,{right arrow over (Δλ₀₁)},{right arrow over (Δλ₀₂)})=M_(LMS->XYZ) {right arrow over (lms)} _(E)(λ,{right arrow over(α₀)},{right arrow over (λ)}₀,{right arrow over (Δλ₀₁)}, {right arrowover (Δλ₀₂)})   Eq. (3-6)

and where

$\begin{matrix}{{\overset{\rightarrow}{a} = \begin{pmatrix}a_{L} \\a_{M} \\a_{S}\end{pmatrix}},{\overset{\rightarrow}{\lambda} = \begin{pmatrix}\lambda_{L} \\\lambda_{M} \\\lambda_{S}\end{pmatrix}},{\overset{\rightarrow}{{\Delta\lambda}_{1}} = \begin{pmatrix}{\Delta\lambda}_{1L} \\{\Delta\lambda}_{1M} \\{\Delta\lambda}_{1S}\end{pmatrix}},{\overset{\rightarrow}{{\Delta\lambda}_{2}} = \begin{pmatrix}{\Delta\lambda}_{2L} \\{\Delta\lambda}_{2M} \\{\Delta\lambda}_{2S}\end{pmatrix}}} & {{Eq}.\mspace{14mu} \left( {3\text{-}7} \right)}\end{matrix}$

The “0” subscript in the parameter vectors {right arrow over(α₀)},{right arrow over (λ₀)},{right arrow over (Δλ₀₁)}, {right arrowover (Δλ₀₂)} above indicate that there are fixed baseline parametersthat provide an optimal match to the CIEXYZ human observer functionsx(λ),y(λ),z(λ). Thus, when the parameter vectors {right arrow over(α)},{right arrow over (λ)},{right arrow over (Δλ₁)},{right arrow over(Δλ₂)} equal the baseline parameter vectors {right arrow over(α₀)},{right arrow over (λ)}₀,{right arrow over (Δλ₀₁)},{right arrowover (Δλ₀₂)}, the {right arrow over (xyz)}_(E)(λ) functions match theCIEXYZ human observer functions x(λ),y(λ),z(λ).

If the “fits” show a significant improvement between predicted match andactual visual match for all data, but still result in an error higherthan expected from experimental noise, progressive refinements can bemade to the modeling of the lms(λ) functions. The number of adjustableparameters should be kept as low as possible while providing an adequateprediction of metameric matches between pairs of colors.

The resulting modified x(λ),y(λ),z(λ) can be characterized as a newdefinition of CIEXYZ or can be characterized as a correctionαx(λ),Δy(λ),Δz(λ) to the existing CIEXYZ standard. The nature of thecorrections are fairly intuitive by mere visual inspection of theexisting l(λ),m(λ),s(λ) functions from which the x(λ),y(λ),z(λ)functions are derived. The adjustments are in the way of adjusting themaximum amplitude, the wavelength where the maximum sensitivity occursfor each l(λ),m(λ),s(λ) peak, the width of each peak on either side ofthe maximum, and the shape of each curve (i.e. steepness at half-max) oneither side of maximum. The mathematical model for defining these keyattributes (amplitude, location of maximum, width and shape on eitherside of maximum) can be improved or modified, but must result in asmooth behavior.

The EdgeXYZ method constructs improved x(λ),y(λ),z(λ) observer functionsthat fulfill the following requirements:

-   -   1) The new x(λ),y(λ),z(λ) average observer functions are based        on available cmf data, such as that of Guild and Wright and/or        Stiles and Burch, as well as recently measured spectra of        visually metameric pairs of colors including whites, as well as        various saturated colors, obtained by Thornton and by Alvin.    -   2) The resulting new x(λ),y(λ),z(λ) are inherently smooth by        means of constraining the number of modeling parameters to be        much smaller than the typical number of wavelength samples (i.e.        much smaller than 3×32 for all the observer functions        x(λ),y(λ),z(λ)), e.g. 3×10 parameters for EdgeXYZ in the example        described in the previous section.    -   3) The resulting new x(λ),y(λ),z(λ) reduces the average ΔE        discrepancy between metameric whites for a statistically        significant population of observers.    -   4) The resulting new x(λ),y(λ),z(λ) also reduces the average ΔE        discrepancy between highly chromatic metameric colors.    -   5) For a population of observers, any remaining errors between        predicted matches based on a new x(λ),y(λ),z(λ) between        numerically metameric colors and visually metameric colors        should be confirmed to be due to random differences and        variability between the observers.

This last requirement means that, for example, given a pair of metamersthat are calculated to have similar values of XYZ using the newx(λ),y(λ),z(λ) functions, there will be a significant percentage ofobservers who say that the two colors are a reasonable match. For thosewho see a difference within that population, the “fix” to thatdifference should be equally distributed as corrections in directions ofred, green, blue.

Any systematic improvements needed to the color match between calculatedmetamers that are consistent in color direction for the test population(for example, on average subjects see too much red in one of the twocolors), should be regarded as an indication of something missing in theobserver functions x(λ),y(λ),z(λ) or something not quite correct in ourfundamental assumptions of how color matching should be predicted. Whileany non-systematic improvements (i.e. improvements equally distributedin all color directions for the population of observers) are likely dueto a random distribution of observer-to-observer differences.

Methods for Improved Precision in the Acquisition of Observer Data

As noted above, the historical experiments for characterizing the CMFsusing the saturation method have significant noise, on the order of 10%-20%. In light of today's knowledge of color appearance modeling, thereare two complimentary experiments that may be performed in order toensure an accurate characterization of the CMFs as well as to ensurethat the revised LMS color responses will result in good predictions ofmatching colors in the vicinity of white. The latter improvement will beof great benefit particularly to the paper industry and to manufacturersof light sources that require tight specifications.

-   -   1) Improved saturation method of acquiring cmfs. The        conventional color matching experiments may be compared to        performing sensitive color adjustments on RGB images on a        wide-gamut display with a gamma=1.0. One can easily recreate the        problems associated with such testing by creating a wide gamut        RGB working space in Adobe PhotoShop™ or other similar program        and setting the value of gamma to 1. Existing images in common        working spaces such as AdobeRGB or sRGB can be converted to the        wide gamut RGB space with gamma=1. If the user proceeds to        adjust colors using the “curve adjust” feature, one finds that        colors shift very gradually for adjustments of R, G or B in the        vicinity of 255, and that colors shift dramatically for        adjustments in the vicinity of R,G, or B=0. This highly        non-uniform response of color appearance to adjustments of RGB        can lead to significant noise when performing human observer        experiments. Thus, a simple modification to the original color        matching experiments is to enable adjustments of quantities of        RGB when matching reference colors such that the adjustments and        the data acquired for those adjustments are performed with a        gamma=2.2 rather than gamma=1.0. After the data has been        acquired, averaged, etc. by multiple observers, the final        results can be converted back to RGB linear space.    -   2) Modified Maxwell method for acquiring cmfs. It is known in        color appearance modeling as well as the experience of graphic        artists that the most sensitive regions of color shift are grays        and skin tones. The former region of color space is well-suited        for defining the LMS functions (and therefore XYZ) with an        expected noise of 1% -2% rather than 10%-20%. This appeared to        be true with at least some of the data acquired by North and        Fairchild using the Maxwell method of matching whites. There are        two modifications to the Maxwell approach that must be added.        The first is to apply non-linear response such as a gamma curve        to the adjustable R, G, B intensities in order to make        adjustments to these intensities resemble the perceptual        response of the eye. The second requirement would be to perform        this experiment in a similar fashion to the historical        experiments such as those of Guild and Wright and Stiles and        Burch. Rather than using only seven discrete monochromatic        wavelengths (per the experiment of North and Fairchild), the        full visible spectrum would be scanned in relatively small        increments such as 5-10 nm.

The noise and variability in the cmf data acquired by North andFairchild seems large when one considers the sensitivity of the eye towhite and gray balance. It would seem that the noise would be due to oneor more of the following:

-   -   1) Lack of skill on the part of observers to find an optimal        match via manual adjustment; and    -   2) Observer-to-observer differences.

It would be helpful to clarify the noise of obtaining cmf data from eachindividual in the experiment compared to the overall variability of thedata. It would also be helpful to perform the following follow-upexperiment: have one observer with demonstrated good color vision aswell as skill in the process of matching colors decide on an optimalmatch for each wavelength λ using the Maxwell method of matching whites.Store the resulting RGB cmf values for each wavelength λ. Next proceedto query each observer for each set of wavelengths λ whether thereference and adjusted whites are a close match. One can even use thatmatch as a starting point from which the observers can deviate ±Δr, Δg,and Δb. If the noise and variability is similar between the standardunbiased experiment and the “Δr, Δg, and Δb” experiment, this would tendto support the assumption that there is significant observer-to-observereffects occurring.

On the other hand, if the resulting new observer cmf data is much moreconsistent and has a much smaller magnitude of noise for all theobservers, we can conclude that lack of observer skill in the ability tofind the optimal match is a significant factor, one that is notnecessarily related to observer-to-observer differences.

The mathematical framework for the Maxwell method is similar to that ofthe saturation method. One begins by stipulating that XYZ of the whitereference spectrum S_(W)(λ′) must be equal to the XYZ due the RGB CRTprimaries plus the XYZ of the monochromatic light of wavelength λ:

$\begin{matrix}\begin{matrix}{{\overset{\rightarrow}{XYZ}\left( {S_{W}\left( \lambda^{\prime} \right)} \right)} = {\overset{\rightarrow}{XYZ}}_{W}} \\{= {{{M_{XYZ}\left( {{\overset{\rightarrow}{S}}_{T}\left( \lambda^{\prime} \right)} \right)}{\overset{\rightarrow}{RGB}\left( {\lambda,{{\overset{\rightarrow}{S}}_{T}\left( \lambda^{\prime} \right)}} \right)}} + {\overset{\rightarrow}{xyz}(\lambda)}}} \\{{\overset{\rightarrow}{RGB}\left( {\lambda,{{\overset{\rightarrow}{S}}_{T}\left( \lambda^{\prime} \right)}} \right)} = {\left( {M_{XYZ}\left( {{\overset{\rightarrow}{S}}_{T}\left( \lambda^{\prime} \right)} \right)} \right)^{- 1}\left( {{\overset{\rightarrow}{XYZ}}_{W} - {\overset{\rightarrow}{xyz}(\lambda)}} \right)}} \\{= {\left( {M_{{LMS}\rightarrow{XYZ}}^{- 1}{M_{LMS}\left( {{\overset{\rightarrow}{S}}_{T}\left( \lambda^{\prime} \right)} \right)}} \right)^{- 1}M_{{LMS}\rightarrow{XYZ}}^{- 1}}} \\{\left( {{\overset{\rightarrow}{LMS}}_{W} - {\overset{\rightarrow}{lms}(\lambda)}} \right)} \\{= {{M_{LMS}^{- 1}\left( {{\overset{\rightarrow}{S}}_{T}\left( \lambda^{\prime} \right)} \right)}{M_{{LMS}\rightarrow{XYZ}}\left( {{\overset{\rightarrow}{LMS}}_{W} - {\overset{\rightarrow}{lms}(\lambda)}} \right)}}} \\{= {{M_{LMS}^{- 1}\left( {{\overset{\rightarrow}{S}}_{T}\left( \lambda^{\prime} \right)} \right)}\left( {{\overset{\rightarrow}{LMS}}_{W} - {\overset{\rightarrow}{lms}(\lambda)}} \right)}}\end{matrix} & \left. {{Eq}.\mspace{14mu} \left( {4\text{-}1} \right)} \right)\end{matrix}$

Similar to the classic CMF experiments, the LMS cone responses can beparameterized and a least square fit performed to the CMF data todetermine the most accurate characterization of LMS.

Improvements to LMS->XYZ

The scientific approach to color science with regards to XYZ hasremained essentially unchanged since the early 1930's and may benefitfrom thinking about the human observer in light of today's technology.Rather than fixing or improving the basis for deriving XYZ from LMS, theapproach has been to create newer, more complex CAMs that begin byconverting XYZ to an LMS-type space using at least two well knowndifferent matrices in the art (Hunt-Pointer-Estevez and Bradford).

Since CIEXYZ and CIELAB continue to be used extensively and are thebuilding blocks for the ICC formats, a good argument can be made forimproving CIEXYZ and CIELAB to the point where the quality of theresults for gamut mapping and chromatic adaptation are comparable to ifnot superior to existing CAMs for many common situations. It would bevery beneficial if the color reproduction of saturated RGB images couldbe considered optimal if the color is reproduced on devices of muchsmaller gamuts.

In particular, just as the standard CIE observer functions are the basisof color management, the approaches used in color management may behelpful in redefining the basis for determining the CIE XYZ humanobserver functions. To do this preserving the term “XYZ” in order toemphasize that if successful, the impact on existing XYZ and CIELABinfrastructures would be minimal. However, the interpretation of themeaning of XYZ will be significantly changed.

A model for the human observer analogous to the construction of adigital camera is first constructed. It is assumed that the eye andbrain comprise of RGB detectors (the cones) with linear responsefunctions that are subsequently processed by circuitry and signalprocessing as illustrative shown in FIG. 11. System 1100 includes anL(λ) receptor 1102, a M(λ) receptor 1104, and an S(λ) receptor 1106,each being amplified by corresponding amplifier 1110-1112 prior to beingpresented to a mixing and post-processing circuit 114. The mixing andpost-processing circuit 114 performs the appropriate processing togenerate the XYZ output 1116.

The term “XYZ” is continued to be used in order to maintain continuitywith the past. However, we will now interpret “XYZ” to indicate thesensation of red, gray, and blue in the brain, as opposed to LMS whichare the sensitivities of the cones in the eye. It already has been shownthat color matching depends only on the LMS wavelength sensitivities ofthe cones in the eye. The fact that equal energy light appears whiteindicates that the integrals of LMS can be set to approximately equalvalues. It is assumed that the response to light striking each cone istruly linear not necessarily in terms of perception, but in terms ofmatching color. For example, if the LMS functions themselves were notlinear in their additivity with regards to matching, one would expect adramatically different amount of power required for the sum of threenarrow bands RGB balanced for white vs. an equal energy spectrum ofwhite light. The fact that the integrals resulting in XYZ for the twocases above are similar in magnitude and correctly predict a good matchwith regards to luminosity confirms that simple additivity holds to atleast a first order approximation.

Next, the interaction at the retina and brain between L, M, and S isconsidered. The matrices for converting XYZ->LMS such as those of Huntimply the conversion from LMS->XYZ. Since it has been confirmed that LMSis the basis for color matching (i.e. LMS may be regarded as the “basisvectors” for color matching), this means that the LMS->XYZ conversion isby definition an LMS “mixing” or “cross-contaminating” transformation.Since performing such a conversion does not impact color matching, whatprecisely is the expected impact of this mixing on a human observer, ifany? Phrased another way, is there any physical justification for thecurrent LMS->XYZ mixing transforms?

There is indeed a common human experience that justifies to some extentthe current mixing or contaminating of LMS into XYZ. It is emphasizedagain that X, Y, and Z truly represent the “bottom line” red, gray, bluesensation experienced by the human observer, which is denoted as RGB.

The evidence for the contamination and stimulation of the red observerfunction X by the blue part of the spectrum which can be easily seen inboth the 2 and 10 degree observers is the existence of “violet” at theshortest wavelength end of the visible spectrum. If no contamination orstimulation were occurring in the red X human observer, the shortestwavelength end of the spectrum would simply get deeper and deeper blueuntil it lost visibility. As one embodiment of the invention it iscontended that the red shift of blue towards violet of short wavelengthsimplies that some contamination of the red observer X is indeedoccurring. Thus, this contamination (as seen above) has no impact onmatching all colors that stimulate the LMS in a similar manner willmatch however, the contamination affects how these matching colors areperceived by the human eye+brain: colors in the region of shortwavelength will appear violet, not merely a deeper blue.

A second manifestation of color mixing LMS->XYZ would be that to theextant the LMS functions are subtracted from each other, one will expectmore acute color differentiation to occur, i.e. colors of different LMSvalues will appear significantly different visually. Likewise, to theextant LMS sensitivities are allowed to cross contaminate one another;one would expect reduced color sensitivity and color differentiation.

As a starting point, it may be assumed that L, M, and S map directly toX,Y and Z (i.e. their primary color identification is defined as red,gray, and blue). It is realized that this is a very crude starting pointsince it implies that all the luminance “gray” intensity Y is basedentirely on the medium wavelength cone response M(λ). However, if themethod of LMS->XYZ optimization described below is robust, the validquantities of LMS that combine to yield the sensation of luminosity Ywill automatically and correctly be determined.

The interpretation of positive and negative values of the matrix thatconverts LMS->XYZ is considered. If negative coefficients are used tomix (for example) M (green cone sensitivity) with X (red colorperceived), this implies a subtraction of the M sensitivity from the Lsensitivity, resulting in more acute differentiation. On the other hand,if a positive coefficient is used to mix S (blue cone sensitivity) withX (red color perceived), then the distinction between red and blue isslightly blurred. Hence, any inter-channel mixing of LMS into XYZ in theway of positive coefficients may be regarded as contamination betweenthe LMS sensitivities as they interact with each other in the eye+brainsystem. Inter-channel mixing of LMS into XYZ comprising of negativecoefficients may be regarded as increased differentiation between theLMS sensitivities of the eye.

Thus, the expected outcome of this mixing of LMS->XYZ is anincrease/decrease in color differentiation with negative/positive valuesof mixing of LMS->XYZ. Since it already has been shown that this mixingdoes not impact the matching of color spectra, we conclude that themanifestation of these positive/negative mixing coefficients are:

-   -   1) The human perception of hues, for example “red” mixing with        blue at the violet end of the spectrum.    -   2) The impact on just noticeable differences (JND) and on        magnitudes of large color differences, which have objective        historical data in the form of MacAdam-type ellipses and color        perception ordering systems such as Munsell.

It is the assumption of the present invention that there is great valuein improving the existing conversion of LMS->XYZ and in the coefficientsused to calculate simple color appearance models such as CIELAB in orderto permit significant improvement to these models. Some of the resultsof these improvements are expected to be:

-   -   1) The magnitudes of red color perceived in areas of blue        spectrum should be correct.    -   2) The consistency of the radii of MacAdam-type ellipses plotted        in the modified CIELAB space should be improved.    -   3) Adjacent colors in color-order systems such as Munsell should        have consistent differences of Euclidean distances in color        space, i.e. pairs of colors that are supposed to be equally        spaced perceptually should have similar ΔE's.        Method for Improved Determination of Post Cone Color Mixing of        LMS with Corresponding Improvement to the XYZ Observer        Functions.

It is predicted that the most obvious current deficiency in thecorrelation of JND Oust noticeable difference) quantified as ΔE inCIELAB and human visual experience can be used to estimate modificationsto the mixture of LMS->XYZ that are currently defined. Similarly, thecomparison of differences in color that are considered to be visuallyequal in magnitude (such as the increments of color defined in a colororder system such as Munsell) can be used to optimize LMS->XYZ.

The conversion of LMS->XYZ is essentially a problem with six variables(i.e. the amount of mixing of two channels with each of the primarychannels). The conversion matrix is constrained by the requirement thatthe values of XYZ are equal for an equal energy spectrum.

$\begin{matrix}{\begin{matrix}{\begin{pmatrix}1 \\1 \\1\end{pmatrix} = {M_{{LMS}\rightarrow{XYZ}}^{\prime}\begin{pmatrix}1 \\1 \\1\end{pmatrix}}} \\{= {\begin{pmatrix}m_{11} & m_{12} & m_{13} \\m_{21} & m_{22} & m_{23} \\m_{31} & m_{32} & m_{33}\end{pmatrix}\begin{pmatrix}1 \\1 \\1\end{pmatrix}}}\end{matrix}{m_{11} = {1 - m_{12} - m_{13}}}{m_{22} = {1 - m_{21} - m_{23}}}{m_{33} = {1 - m_{31} - m_{32}}}{M_{{LMS}\rightarrow{XYZ}}^{\prime} = \begin{pmatrix}{1 - m_{XM} - m_{XS}} & m_{XM} & m_{XS} \\m_{YL} & {1 - m_{YL} - m_{YS}} & m_{YS} \\m_{ZL} & m_{ZM} & {1 - m_{ZL} - m_{ZM}}\end{pmatrix}}} & \left( {5\text{-}1} \right)\end{matrix}$

The subscripts i, j in the element m_(i,j) indicate the mixing of L, M,and S into channels X, Y, and Z, i.e. “m_(XM)” indicates the mixing ofmedium sensitivity cone M into the X “red” sensation in the brain.

The improved determination of the amount of positive/negativeinter-channel mixing can be accomplished by defining a variable functionconverting LMS->XYZ->CIELAB via an adjustable LMS->XYZ matrix. Thisadjustable version of CIELAB can be used to calculate the radii of theellipses created from MacAdam-type data sets in the modified CIELABspace and the delta E differences adjacent colors defined in theM-unsell color ordering system as well as any other ordered colorsystems.

In order to calibrate the calculations for visual differences, it willbe assumed that the simplest metric for the most simple range of colorsis correct, i.e. L* for a series of white/gray/black colors. Since thematrix defined above is invariant with regards to white, the values ofL* for neutral colors will not change with optimization of the matrix.Hence, the interpretation of “1 ΔE” with regards to magnitude of visualdifference in color will be the magnitude of visual difference inwhite/gray/black. Differences of 1 ΔE for all other pairs of colors willbe compared to this reference, i.e. is the visual difference between anyparticular pair of colors that differ by 1 ΔE more or less noticeablethan that of white/gray/black. Similar comparisons can be made betweenpairs of colors differing by 5 or 10 ΔE.

The error minimization to be performed will be in the form of minimizingthe difference of the ΔE differences between multiple pairs of colorsdetermined to have a similar magnitude of JND. For example, severalpairs of colors can be extracted from the MacAdam ellipses. Each pairwould comprise of the centroid of one MacAdam ellipse paired with onecolor on the surface of the ellipse. Let i,j denote ellipse i (from theset of N_(ME) MacAdam ellipses) with tristimulous value XYZ_(i0) at thecenter of the ellipse and color sample j (from the set of N_(s) sampleson the ellipse surface) with tristimulous value XYZ_(ij). The firstsummation of squared errors to be minimized will be the differencesbetween the values of ΔE_(ij) for each pair of colors denoted by ij andthe average value ΔE_(ave) for all the pairs:

$\begin{matrix}{{{{\Delta \; {E_{ij}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}} = {{{\overset{\rightarrow}{Lab}}_{ij} - {\overset{\rightarrow}{Lab}}_{i\; 0}}}}{where}{{\overset{\rightarrow}{Lab}}_{ij} = {\overset{\rightarrow}{Lab}\left( {M_{{LMS}\rightarrow{XYZ}}^{\prime}M_{{LMS}\rightarrow{XYZ}}^{- 1}{\overset{\rightarrow}{XYZ}}_{ij}} \right)}}{{\overset{\rightarrow}{Lab}}_{i\; 0} = {\overset{\rightarrow}{Lab}\left( {M_{{LMS}\rightarrow{XYZ}}^{\prime}M_{{LMS}\rightarrow{XYZ}}^{- 1}{\overset{\rightarrow}{XYZ}}_{i\; 0}} \right)}}{{SumSquErr}_{JND}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}} = {{\sum\limits_{{i = 1},{j = 1}}^{{i = N_{ME}},{j = N_{S}}}{\left\lbrack {{\Delta \; {E_{ij}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}} - {\Delta \; {E_{ave}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}}} \right\rbrack^{2}\Delta \; {E_{ave}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}}} = {\frac{1}{N_{ME}N_{S}}{\sum\limits_{{i = 1},{j = 1}}^{{i = N_{ME}},{j = N_{S}}}{\Delta \; {E_{ij}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}}}}}} & {{Eq}.\mspace{14mu} \left( {5\text{-}2} \right)}\end{matrix}$

If perfectly optimized, all the values of ΔE_(ij) would be the same, allΔE_(ij)=ΔE_(ave), and the summation SumSquErr_(JND)(M′_(LMS->XYZ)) wouldbe zero. In reality, due to the noise and uncertainty in the dataacquired by MacAdam, this error function will be minimized, but notnecessarily to 0. The next summation will be for a color-order system(COS) such as Munsell. Based on descriptions of the intent of theMunsell system, the following assumptions may be made for purposes ofoptimizing the LMS->XYZ matrix in order to create a perceptually uniformspace:

-   -   1) Pairs of colors having Munsel value i and i+1 should have the        same change in L* as the corresponding pair of Munsell gray        colors of values i and i+1;    -   2) Pairs of color having the same Munsel hue k and value i, and        chromas j and j+1 should all have the same value of ΔC*; and    -   3) Pairs of color having the same value i, and chroma j, with        hues k and k+1, should differ by the same value of        ΔE_(hue)=(ΔE²-ΔL*²−ΔC*²)^(1/2).        First ΔE, ΔL*, ΔC*, ΔH* are defined between pairs of colors        using the increments Δi, Δj, Δk for the values of value, chroma,        and hue:

$\begin{matrix}\begin{matrix}{{\Delta \; {E_{ijk}\left( {M_{{LMS}\rightarrow{XYZ}}^{\prime},{\Delta \; i},{\Delta \; j},{\Delta \; k}} \right)}} = {{\overset{\rightarrow}{{Lab}}\left( {\overset{\rightarrow}{XYZ}}_{{i + {\Delta \; i}},{j + {\Delta \; j}},{k + {\Delta \; k}}}^{\prime} \right)} -}} \\{{\overset{\rightarrow}{Lab}\left( {\overset{\rightarrow}{XYZ}}_{ijk}^{\prime} \right)}} \\{{\Delta \; {L_{ijk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}} = {{L^{*}\left( {\overset{\rightarrow}{XYZ}}_{{i + 1},j,k}^{\prime} \right)} -}} \\{{L^{*}\left( {\overset{\rightarrow}{XYZ}}_{i,j,k}^{\prime} \right)}} \\{{\Delta \; {C_{ijk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}} = {{C^{*}\left( {\overset{\rightarrow}{XYZ}}_{i,{j + 1},k}^{\prime} \right)} -}} \\{{C^{*}\left( {\overset{\rightarrow}{XYZ}}_{i,j,k}^{\prime} \right)}} \\{\left\lbrack {\Delta \; {H_{ijk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}} \right\rbrack^{2} = {\left\lbrack {\Delta \; {E_{ijk}\left( {M_{{LMS}\rightarrow{XYZ}}^{\prime},{0,0,1}} \right)}} \right\rbrack^{2} -}} \\{\left\lbrack {{L^{*}\left( {\overset{\rightarrow}{XYZ}}_{i,j,{k + 1}}^{\prime} \right)} -} \right.} \\{\left. {L^{*}\left( {\overset{\rightarrow}{XYZ}}_{i,j,k}^{\prime} \right)} \right\rbrack^{2} -} \\{\left\lbrack {{C^{*}\left( {\overset{\rightarrow}{XYZ}}_{i,j,{k + 1}}^{\prime} \right)} -} \right.} \\\left. {C^{*}\left( {\overset{\rightarrow}{XYZ}}_{i,j,k}^{\prime} \right)} \right\rbrack^{2}\end{matrix} & {{Eq}.\mspace{14mu} \left( {5\text{-}3} \right)}\end{matrix}$

where

{right arrow over (XYZ)}′=M′ _(LMS->XYZ) M ⁻¹ _(LMS->XYZ) {right arrowover (XYZ)}  Eq. (5-4)

Next the sum squared error of the differences of the ΔE differences fromthe average ΔE within a series of colors that are determined to beequally spaced perceptually are determined. The sums are separatedaccording to colors equally spaced in value (ΔL*), chroma (ΔC*), and hue(ΔE_(hue)):

$\begin{matrix}\begin{matrix}{{{SumSquErr}_{L^{*}}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} = {\sum\limits_{{i = 1},{j = 1},{k = 1}}^{{i = N_{Value}}{j = N_{Chroma}}{k = N_{Hue}}}\left\lbrack {{\Delta \; {L_{ijk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}} -} \right.}} \\\left. {{\overset{\_}{\Delta \; L}}_{jk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} \right\rbrack^{2} \\{{{\overset{\_}{\Delta \; L}}_{jk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} = {\frac{1}{N_{Value}}{\sum\limits_{i = 1}^{i = N_{Value}}{\Delta \; {L_{ijk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}}}}}\end{matrix} & {{Eq}.\mspace{14mu} \left( {5\text{-}5} \right)} \\\begin{matrix}{{{SumSquErr}_{C^{*}}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} = {\sum\limits_{{i = 1},{j = 1},{k = 1}}^{{i = N_{Value}}{j = N_{Chroma}}{k = N_{Hue}}}\left\lbrack {{\Delta \; {C_{ijk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}} -} \right.}} \\\left. {{\overset{\_}{\Delta \; C}}_{ik}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} \right\rbrack^{2} \\{{{\overset{\_}{\Delta \; C}}_{{ik}\; {Chroma}}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} = {\frac{1}{N_{Chroma}}{\sum\limits_{j = 1}^{j = N_{Value}}{\Delta \; {C_{ijk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}}}}}\end{matrix} & {{Eq}.\mspace{14mu} \left( {5\text{-}6} \right)} \\\begin{matrix}{{{SumSquErr}_{H^{*}}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} = {\sum\limits_{{i = 1},{j = 1},{k = 1}}^{{i = N_{Value}}{j = N_{Chroma}}{k = N_{Hue}}}\left\lbrack {{\Delta \; {H_{ijk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}} -} \right.}} \\\left. {{\overset{\_}{\Delta \; H}}_{ij}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} \right\rbrack^{2} \\{{{\overset{\_}{\Delta \; H}}_{ij}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} = {\frac{1}{N_{Hue}}{\sum\limits_{k = 1}^{k = N_{Hue}}{\Delta \; {H_{ijk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}}}}} \\{{{SumSquErr}_{C^{*}}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} = {\sum\limits_{{i = 1},{j = 1},{k = 1}}^{{i = N_{Value}}{j = N_{Chroma}}{k = N_{Hue}}}\left\lbrack {{\Delta \; {C_{ijk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}} -} \right.}} \\\left. {{\overset{\_}{\Delta \; C}}_{ik}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} \right\rbrack^{2} \\{{{\overset{\_}{\Delta \; C}}_{{ik}\; {Chroma}}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} = {\frac{1}{N_{Chroma}}{\sum\limits_{j = 1}^{j = N_{Value}}{\Delta \; {C_{ijk}^{*}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}}}}}\end{matrix} & {{Eq}.\mspace{14mu} \left( {5\text{-}7} \right)}\end{matrix}$

It is noted that the values of N_(Value), N_(Chroma), and N_(Hue) areactually not fixed throughout color-order systems such as Munsell, butrather vary as a function of i, j, and k. For example, the value ofN_(value) is 10 for a gray scale, (Chroma=0), but may be 1 or 2 forsaturated regions of color space, i.e. colors with large value ofchroma. Thus the values of N_(Value) should be dynamically calculated ordetermined for a particular set of values for Chroma and Hue (j and k).It is assumed for simplicity of documentation that the above summationsimplicitly contain values of N_(Value), N_(Chroma), and N_(Hue) thatdepend on i and/or j and/or k.

The above are combined into one summation that captures the totaldiscrepancy or error in perceptual uniformity between CIELAB and acolor-order system such as Munsell. This is the error function to beminimized via optimization of the LMS->XYZ matrix:

$\begin{matrix}{{{SumSquErr}_{COS}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} = {{{SumSquErr}_{L^{*}}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} + {{SumSquErr}_{C^{*}}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)} + {{SumSquErr}_{H^{*}}\left( M_{{LMS}\rightarrow{XYZ}}^{\prime} \right)}}} & {{Eq}.\mspace{14mu} \left( {5\text{-}8} \right)}\end{matrix}$

The two summations, JND and COS can be combined into a total error to beminimized:

SumSquErr_(Total)(M′ _(LMS->XYZ))=SumSquErr_(JND)(M′_(LMS->XYZ))+SumSquErr_(COS)(M′ _(LMS->XYZ))   Eq.(5-9)

This ensures that the resulting optimized matrix yields the closestagreement possible with these two very different historicalcharacterizations of perceptual uniformity—MacAdam ellipses (or similarJND data) and color-order systems such as Munsell.

The coefficients contained in M′_(LMS-XYZ) can be shown to have a largeimpact on the symmetry and shapes of contour plots of constant Munsellchroma at constant value (i.e. L*). The diameters of the contours in thea* and b* directions are defined by the coefficients contained in theexpressions for a* and b* in CIELAB.

Since the above optimizations of LMS->XYZ will be affected by thecoefficients that currently exist in the equations for scaling L*a*b*,the coefficients for a* and b* will be added to the optimizationprocess.

A least squares fit (LSF) was performed on the Munsell data set acquiredfrom RIT entitled “real.dat”. This data was apparently measured withnear-D65 illumination. All measured values of Yxy provided by RIT, aswell as the near-D65 Yxy value of illumination were converted to XYZ andfrom XYZ to LMS via the Hunt-Pointer-Estevez matrix from section 6 ofthis report. These values of LMS were converted to EdgeXYZ by means ofan adjustable LMS->XYZ matrix as indicated in Equation 1-12 above. Next,the values of EdgeXYZ were converted to EdgeLAB using the recalculatedvalue of EdgeXYZ white reference for the near-D65 illumination, andusing adjustable values of the coefficients for a* and for b*.

In the fit, the values of m_(ij) were automatically adjusted as well asthe scaling coefficients used to calculate a* and b* (currently 500 and200 respectively in the CIELAB equations). The error minimization wasperformed using Powell's method in one embodiment, using the averagechange in chroma of CIELAB between increments of Munsell chroma forcalculating the standard deviation in chroma increments for EdgeLAB.This latter choice was made to ensure that EdgeLAB would be asconsistent as possible with historical metrics, and to avoid reducingthe standard deviation of the chroma increments by accidentally reducingthe overall chroma itself for all colors.

The quality metric for assessment was to calculate the standarddeviation for equal steps of L* (value), equal steps of hue, and equalsteps of chroma. The following parameters resulted from this errorminimization:

TABLE 12 Calculated Parameters for EdgeXYZ LMStoXYZMatrix LMS_L LMS_MLMS_S XYZ_X = 1.83489 −1.0069 0.17201 XYZ_Y = 0.282207 0.733972−0.0161793 XYZ_Z = −0.0508106 0.162992 0.887818 a * Coefficient = 524.90b * Coefficient = 220.12 mSigmaEdgeL = 0.29 mSigmaEdgeC = 1.08mSigmaEdgeH = 1.28 mSigmaCIE_L = 0.25 mSigmaCIE_C = 1.54 mSigmaCIE_H =1.34As can be seen, the above modified values of the LMS->XYZ matrix andcorresponding coefficients for a*b* in CIELAB result in a 33%improvement in the consistency of the calculated deltaE in the directionof chroma as defined in the Munsell Color Order System. One notes thatthe biggest impact of optimizing the LMS->XYZ matrix is on the symmetryof contour plots of equal chroma at different fixed levels of value. Thelop-sided contours in the direction of yellow for CIELAB appear to implycontamination of the blue response of the eye+brain, Z(λ).

A test was performed to confirm the validity of the Munsell color ordersystem. Although the optimization process resulted in changes to CIELAB,the changes were not dramatic since CIELAB had already been optimized toagree well with Munsell. If the assumptions regarding how Munsell colorsshould be interpreted were incorrect, the entire basis upon which CIELABwas optimized could be false.

In the test, two charts were created. In the first chart, steps of colorwere created that differed from their surround by 10, 20, and 30 delta Eunits in the direction of chroma for each of the major 6 directions ofcolor (Y, R, M, B, C, G) for three levels of L*—60, 50, and 40. In thesecond chart, steps of color were created that differed from theirsurround by 10, 20, and 30 delta E units in the direction of reducedchroma, direction of reduced L*, and in a counterclockwise direction ofhue for a single value L*=70 for the surround, again replicated for eachof the 6 major directions of color.

It was clear that both CIEXYZ and CIELAB as well as EdgeXYZ and EdgeLABas defined using the Munsell data were inaccurate by a significantamount, implying that either Munsell was flawed or the interpretationfor Munsell was flawed that was used to optimize CIELAB. The magnitudeof visual difference for changes in yellow chroma compared to thechromas of other colors appeared to be inaccurate by a factor of 1.5.Likewise, the magnitude of visual difference for changes in L* for allthe colors appeared to be larger by a factor of 4 than comparablechanges in chroma and hue.

It was also observed that in many colors, as chroma increased at fixedL*, the visual luminosity increased by approximately 3 delta E units inL* for a 30 delta E change in chroma. This phenomenon has been observedfor many years and is called the Helmholtz-Kohlrausch Effect (seesection 6-5 of Fairchild's Color Appearance Models). Typically,empirical corrections were derived that added complexity to colorappearance models.

However, with the interpretation described above for the relationship ofLMS->XYZ, such a phenomenon has a simple explanation. If it appears thatfor example steps of red increase in perceived L* as chroma increases,this clearly implies that the quantity of long wavelength cone responseL(λ) mixed in to define luminosity Y (i.e. the value of m_(YL) in theLMS->XYZ matrix) is too small. Increasing this value will result inlarger values of Y as the magnitude of cone response L(λ) increasesrelative to cone response M(λ).

Since the Munsell-based optimization of EdgeXYZ and EdgeLAB did notresult in a satisfactory test, the two charts described above wereadjusted manually in order to ensure that the differences in alldirections of color resulted in a similar magnitude of visualdifference. This data was then used to improve LMS->XYZ matrix and thea*b* coefficients through manual adjustment of the parameters. It wasfound that it was relatively easy to determine improved values thatagreed with the values in the modified charts to within the uncertaintyof the visual adjustments performed in order to optimize the appearanceof the charts (i.e. less than ±5 deltaE in units of CIELAB). Thequalitative modifications to both the LMS->XYZ matrix as well as to thea* and b* coefficients was quite significant, as indicated below inTable 13:

TABLE 13 L M S X 1.9202 −1.1121 0.1919 Y 0.6210 0.3390 0.0400 Z 0.35000.1500 0.5000 EdgeCoeff_a 130.00 EdgeCoeff_b 80.00

Qualitatively, the big differences from the Munsell-based optimizationare:

-   -   1) Significant contamination of blue channel Z with “red” and        “green” cone response (L and M)    -   2) Significant increase in red impact on luminosity or gray        sensation “Y”    -   3) Reduction of coefficients for “a*” and “b*” by a factor of 3    -   4) Reduction of the ratio of coefficients for a* and b* from 5/2        (=2.5) to 13/8 (=1.625)

Based on the above, a new color order system can be defined with similarconcepts of lightness (value), hue, and chroma as Munsell. This newcolor order system can be laid out in units of 10 deltaE for L* and C*.The identification of hues would be similar to Munsell.

It appears that significant improvements can be made both to colormatching and color perception. The former improvement entails refiningthe definition of the LMS cone sensitivity functions, the latter entailsoptimizing the conversion of LMS->XYZ and the coefficients for a* and b*in order to correlate well with color perception, both JND andcolor-order systems. Using validation tests to confirm perceptualconsistency, color order systems themselves can be confirmed orsignificantly improved by using the results of the tests to generatesignificantly better versions of EdgeXYZ and EdgeLAB.

The above described method for improvement to the LMS cone responseswill have some of the following impact in some embodiments of theinvention:

-   -   1) Reconcile the 25 ΔE discrepancy in the matching of whites        with significantly different spectra;    -   2) Reconcile errors in matching saturated colors;    -   3) Should enable excellent visual matches between images on any        display and corresponding hard copy images under an illuminant,        or backlit transparencies;    -   4) Enable accurate matches between sources of illumination such        as standard fluorescent tubes even when spectra are different;    -   5) Enable good visual match between images digitally projected        with either projected film images or original colors in a scene;        and    -   6) Allow accurate color between flat panel displays, HDTV, etc.        between display venders and between displays and other forms of        color imaging such as printed advertisements and packaging.

The above described method for improvement to LMS->XYZ and thecoefficients of a* and b* in EdgeLAB will have a large impact on:

-   -   1) Quality of color reproduction when mapping from large gamut        to smaller gamut systems due to having a more uniform perceptual        color space;    -   2) Optimizing the appearance of saturated spot colors printed on        devices with moderate color gamut such as ink jet; and    -   3) Quantifying the magnitude of ΔE errors for specifying inks,        display colorants, and proofing systems.

Referring now to FIG. 12, there is shown a block diagram illustrating anexemplary operating environment for application of the technique(s)previously discussed above in accordance with an embodiment of theinvention. In particular, FIG. 12 shows a general purpose-computingenvironment or system 1200 comprising a processor 1202, memory 1204,user interface 1206, source-imaging device 1208 and destination-imagingdevice 1210.

Source imaging device 1208 and destination imaging device 1210 may bedisplay devices or printers in one embodiment. Source imaging device canact as an input for spectral information into processor 1202.Source-imaging device 2108 may be an image capture device such as ascanner or camera. In either case, source imaging device 1208 anddestination imaging device 1210 operate according to an applicable setof device-dependent coordinates. As an example, source-imaging device1208 may be an image capture device that produces source device datadefined by RGB coordinates, while destination-imaging device 1210 may bea printer that produces destination device data. Memory 1204 can includeboth volatile (e.g., RAM) and nonvolatile (e.g., ROM) storage and canstore all of the previously discussed characterization and correctiontechniques.

User interface 1206 may include a display device, such as a cathode raytube (CRT), liquid crystal display (LCD), plasma display, digital lightprocessing (DLP) display, or the like, for presentation of output to auser. In addition, user interface 1206 may include a keyboard andpointing device, such as a mouse, trackball or the like to support agraphical user interface. In operation, a user interacts with processor1202 via user interface 1206 to direct conversion of source device datato destination device data. Memory 1204 stores process-readableinstructions and data for execution by processor 1202. The instructionsinclude program code that implement the chromatic adaptation techniquesdescribed herein. Processor 1202 may operate in any of a variety ofoperating systems, such as Windows™, Mac OS, Linux, Unix, etc.

Processor 1202 may take the form of one or more general-purposemicroprocessors or microcontrollers, e.g., within a PC or MAC computeror workstation. In an alternative embodiment, the processor 2102 maybeimplemented using one or more digital signal processors (DSPs),application specific integrated circuits (ASICs), field programmablelogic arrays (FPGAs), or any equivalent integrated or discrete logiccircuitry, or any combination thereof. Memory 2104 may include orutilize magnetic or optical tape or disks, solid state volatile ornon-volatile memory, including random access memory (RAM), read onlymemory (ROM), electronically programmable memory (EPROM or EEPROM), orflash memory, as well as other volatile or non-volatile memory or datastorage media. Processor 1202 can execute the techniques disclosedherein for color measurement and other methods using programs storedwithin system 1200 or outside of system 1200 (i.e., stored in anexternal database, etc.).

FIG. 13 shows a graph highlighting a comparison of the current inventionXYZ versus CIELAB XYZ in accordance with an embodiment of the invention.FIG. 14 highlights a device such as a digital camera or colorimeter 1400that includes an input 1402 for receiving a color stimulant; one or morefilters 1404 optimized for filtering the color stimulant for determiningEdgeLMS and EdgeXYZ from the color stimulant without requiring a fillspectral measurement, the filtering. Filters 1404 are preferablyexecuted using controller 1408. The filters 1404 are preferably softwarefilters that determine EdgeLMS and EdgeXYZ and provided at output 1406as described herein. This provides for a very efficient way ofdetermining the required measurement without having to resort to a fullspectral measurement which is time consuming.

In FIG. 15, there is shown a flowchart highlighting a method fordetermining color matching functions in accordance with an embodiment ofthe invention. In 1502, spectral data from metameric pairs is obtained.In 1504, new color matching functions are generated by modifyingoriginal color matching functions. While in 1506, the new color matchingfunctions are constrained to the original color matching functions whilereducing calculated perceptual error between the metameric pairs. In oneembodiment of the invention, calculated perceptual error is reduced byat least 50 percent.

In FIG. 16, a flowchart highlighting a method for determining colormatching functions in accordance with an embodiment of the invention ishighlighted. In 1602, spectral data originating from color matchingexperiments and from metameric pairs is obtained. Color matchingfunctions from a set of parameters is defined in 1604. In 1606, an errorfunction that indicates error due to perceptual differences between theparameterized color matching functions and the color matchingexperiments is defined. In one embodiment the number of parameters inthe set of parameters is set to less than 96.

Some further embodiments of the invention will be now described. Theseembodiments are not intended to be limited but are described in order toprovide a better understanding of some of the uses and differentapplications for the invention. In one embodiment, a method foroptimizing the EdgeLMS cone response functions, the conversion ofEdgeLMS->EdgeXYZ, and the calculation of EdgeLAB from EdgeXYZ isaccomplished. The method entails optimizing parameters in order toachieve good agreement between the metrics used to define color and avariety of experimental data based on human observers. The optimizationcan be performed manually via trial and error or can be performedautomatically via least squares fit or other well known techniques.Another embodiment of the invention comprises an apparatus that acceptsa variety of experimental data based on human observers andautomatically generates the optimal modifications to EdgeLMS, EdgeXYZ,and EdgeLAB in the manner described above.

Another embodiment of the invention comprises an apparatus that containsthe improved characterizations of EdgeLMS, EdgeXYZ, and/or EdgeLAB thathave already been determined via the method or apparatus above. Theinput to this apparatus includes spectral measurement data, with theoptional spectral measurement of a white reference and/or whiteilluminant. The output of the apparatus is the calculated EdgeLMS and/orEdgeXYZ. If spectral white reference data is included, the apparatus canalso output values of EdgeLAB. The value of this apparatus is that anytwo colors that have the same EdgeXYZ and/or EdgeLAB will be a goodvisual match to one another even if their measured spectra aresignificantly different. Note that the apparatus may be contained withinthe measurement device itself without the need for an external computerto provide a convenient way to measure and display the values ofEdgeLMS, EdgeXYZ, or EdgeLAB.

In still another embodiment of the invention, an apparatus that containsthe previously determined definitions of EdgeLMS, EdgeXYZ, and EdgeLABand that receives as input spectral data files corresponding to avariety of color measurements. The output of the apparatus includes anICC profile, or equivalent thereof, based upon EdgeLMS, EdgeXYZ, and/orEdgeLAB. Alternatively, the apparatus may receive data files thatalready contain the calculated values of EdgeLMS, EdgeXYZ, and/orEdgeLAB for purposes of creating an ICC profile or the equivalentthereof. Such an apparatus would optionally have a means of identifyingthat the ICC profile thus generated is based upon the Edge colorstandard vs. the CIE standard as currently defined in the CIE 2 or 10degree observers.

Another embodiment of the invention comprises an apparatus that receivesas input CIE color data and/or ICC profiles based on CIE color data. Theapparatus performs all color mapping from source to destination devices,particularly for out of gamut colors, by converting CIELAB->EdgeLAB andEdgeLAB->CIELAB for the source and destination profiles in order toobtain optimal gamut mapping. If the input is spectral data, thenEdgeLMS can likewise be calculated as well for optimal color matching.The output of such an apparatus is either a device link for convertingimage data from a source device to a destination device or the convertedimage itself if one of the inputs is image data to be converted fromsource to destination device.

Another embodiment of the invention is an apparatus that capturesdigital color images, prints digital color images, displays digitalcolor images, or otherwise processes digital color images. The apparatuscontains color characterization information for the device either in theform of an ICC profile or equivalent thereof, or in the form ofessential colorimetric data such as RGB chromaticities and white pointin the example of an RGB display. The apparatus contains thecharacterization information utilizing the previously optimized EdgeLMS,EdgeXYZ, and/or EdgeLAB. The output of the device is either digitalcolor data or a color image that is ensured to be visually accurate byutilizing the characterization information for the device based upon anapparatus that receives as input requests for defining either a sourceor destination device or both based upon a previously determinedstandard such as SWOP_C3 (defined by IDEALliance, international digitalenterprise alliance) or AdobeRGB (defined by Adobe, Inc.) or any otherapplicable standard wherein the apparatus utilizes the Edge colorstandard to define the standard device characterization and to performconversions of image color data.

The invention has been described in detail with particular reference tocertain preferred embodiments thereof, but it will be understood thatvariations and modifications can be effected within the scope of theinvention.

1. A method for determining color matching functions, comprising:obtaining spectral data originating from metameric pairs; generating newcolor matching functions by modifying original color matching functions;and wherein the new color matching functions are constrained to besimilar to the original color matching functions while reducingcalculated perceptual error between the metameric pairs by at least 50percent.
 2. A method as defined in claim 1, wherein a perceptual errorbetween the metameric pairs is reduced by at least a factor of two.
 3. Amethod as defined in claim 1, further comprising: constraining the newcolor matching functions to have a smooth behavior; and using the newcolor matching functions to generate a profile connecting space (PCS)for converting colors.
 4. A method as defined in claim 1, furthercomprising: using the new color matching functions for generatingchromaticities that are used for defining a standard RGB space.
 5. Amethod as defined in claim 1, further comprising: using the new colormatching functions for characterizing a display.
 6. A method as definedin claim 1, further comprising: using the new color matching functionsfor defining paint colors.
 7. A method as defined in claim 6, whereinstandard values defined for matching the paint colors are based on thenew color matching functions.
 8. A method as defined in claim 1, furthercomprising: using the new color matching functions for soft proofingwherein a profile connecting space (PCS) between color profiles for ahard copy proof and a soft proof is based on the new color matchingfunctions.
 9. A method as defined in claim 1, further comprising: usingthe new color matching functions in a color management system wherein aprofile connecting space (PCS) used in the color management system isbased on the new color matching functions.
 10. A method for determiningcolor matching functions, comprising: obtaining spectral dataoriginating from color matching experiments and from metameric pairs;defining color matching functions from a set of parameters; and definingan error function that indicates error due to perceptual differencesbetween the parameterized color matching functions and the colormatching experiments, wherein the number of parameters is less than 96.11. A method as defined in claim 10, further comprising: defining asecond error function that indicates error due to spectral differencesbetween the parameterized color matching functions and the colormatching experiments.
 12. A method as defined in claim 11, furthercomprising: defining a third error function that indicates error due toperceptual differences between the metameric pairs as calculated usingthe parameterized color matching functions.
 13. A method as defined inclaim 10, further comprising: generating new color matching functions byadjusting parameters of the parameterized color matching functions suchthat the error function is minimized.
 14. A method as defined in claim13, wherein resulting average and maximum perceptual errors calculatedwith the new color matching functions are reduced by a predeterminedamount compared to the color matching functions derived from the colormatching experiments.
 15. A method as defined in claim 14, wherein thepredetermined amount that the perceptual errors are reduced comprises atleast a factor of two.
 16. A method as defined in claim 13, furthercomprising: constraining the new color matching functions to have asmooth behavior; generating a profile connecting space (PCS) forconverting colors using the new color matching functions; andcharacterizing a display using the new color matching functions.
 17. Amethod for optimizing the definitions of LMS cone response functions,comprising: defining parameters and parametric equations that aresubstantially similar to current definitions for LMS cone responsefunctions based on the CIE XYZ observer functions; and adjusting theparameters to maintain the substantial similarity to the currentdefinitions for the LMS cone response functions while reducing thecalculated delta E (ΔE) errors between matching pairs of metamericcolors.
 18. A method for optimizing the definitions of XYZ humanobserver functions, comprising: defining parameters and parametricequations that are substantially similar to current definitions for LMScone response functions based on the CIE XYZ observer functions;converting the parameters and parametric equations to XYZ human observerfunctions; adjusting the parameters to maintain the similarity to thecurrent definitions for the LMS cone response functions; and adjustingthe parameters and parametric equations to maintain substantialsimilarity to the XYZ human observer functions while reducing thecalculated delta E (ΔE) errors between matching pairs of metamericfunctions.
 19. A method for defining improved definitions of CIELAB,comprising: (a) defining parameters and parametric equations that aresubstantially similar to current definitions for LMS cone responsefunctions based on CIE XYZ observer functions; (b) converting theparameters and parametric equations to XYZ human observer functions; and(c) substituting values in the CIE XYZ observer functions for CIELABwith values of XYZ human observer functions from (b).
 20. A system,comprising: an input for receiving measured spectral data for at leastone color; and a controller coupled to the input for calculating XYZhuman observer defining parameters and parametric equations that aresubstantially similar to current definitions for LMS cone responsefunctions based on the CIE XYZ observer functions and converting theparameters and parametric equations to XYZ human observer functions,adjusting the parameters to maintain the similarity to the currentdefinitions for the LMS cone response functions, and adjusting theparameters and parametric equations to maintain substantial similarityto the XYZ human observer functions while reducing the calculated deltaE (ΔE) errors between matching pairs of metameric functions
 21. A systemas defined in claim 20, wherein the input receives spectral datadefining a reference white, and the controller calculates XYZ humanobserver functions for the reference white and calculates CIELAB colorspace values using the values of XYZ for the at least one color and XYZfor the reference white.
 22. A system, comprising: means for receivingmeasured spectral data associated with a list of device code values;means for converting the spectral data to XYZ or LAB data by: (a)defining parameters and parametric equations that are substantiallysimilar to current definitions for LMS cone response functions based onCIE XYZ observer functions; (b) converting the parameters and parametricequations to XYZ human observer functions; (c) substituting values inthe CIE XYZ observer functions for CIELAB with values of XYZ humanobserver functions from (b); and means for generating an ICC profileusing the XYZ or LAB data.
 23. A system as defined in claim 22, whereinthe ICC profile that is generated is tagged in order to indicate the XYZor LAB metric used to generate the profile.
 24. A method for defininghuman observer functions, comprising: obtaining a set of measuredspectra from a set of color samples; parameterizing a conversion fromcone response functions to human observer functions; converting from thehuman observer functions to a perceptually uniform space; and optimizingthe parameters obtained during the parameterizing in order to obtainimproved correspondence between Euclidean differences and magnitudes ofvisual difference between the color samples.
 25. A device, comprising:an input for receiving a color stimulant; and one or more filtersoptimized for filtering the color stimulant for determining EdgeLMS andEdgeXYZ from the color stimulant without requiring a full spectralmeasurement.
 26. A method for optimizing the coefficients used inconversion from LMS to XYZ, comprising: obtaining measured color datafrom sequences of colors wherein the visual differences between colorswithin each sequence is substantially similar; calculating a delta E(ΔE) difference between adjacent pair of colors within each sequence;calculating the differences in delta E (ΔE) between each pair ofadjacent colors within each sequence; summing the differences in delta E(ΔE) in order to obtain an error function; and optimizing thecoefficients used in the conversion of LMS to XYZ in order to minimizethe error function.
 27. A method for optimizing coefficients used forcalculating CIELAB, comprising: obtaining measured color data fromsequences of colors wherein the visual differences between colors withineach sequence is substantially similar; calculating a delta E (ΔE)difference between adjacent pair of colors within each sequence;calculating the differences in delta E (ΔE) between each pair ofadjacent colors within each sequence; summing the differences in delta E(ΔE) in order to obtain an error function; and optimizing thecoefficients used in the calculation of CIELAB in order to minimize theerror function.
 28. A method for optimizing the coefficients used inconversion from LMS to XYZ and the coefficients used for calculatingCIELAB, comprising: obtaining measured color data from sequences ofcolors wherein the visual differences between colors within eachsequence is substantially similar; calculating a deltaE (ΔE) differencebetween each adjacent pair of colors within each sequence; calculatingthe differences in deltaE (ΔE) between each pair of adjacent colorswithin each sequence; summing the differences in deltaE (ΔE) in order toobtain an error function; and optimizing the coefficients used inconversion from LMS to XYZ and in the calculation of CIELAB in order tominimize the error function.
 29. A method for defining a color ordersystem, comprising: obtaining measured color data from sequences ofcolors wherein the visual differences between colors within eachsequence is substantially similar; calculating a deltaE (ΔE) differencebetween each adjacent pair of colors within each sequence; defining thesequences of colors in directions of lightness, hue, and chroma; andwherein adjacent colors within each sequence differ by the same value ofcalculated deltaE (ΔE).